Colin Maclaurin

How to use this course

Most students meet Colin Maclaurin once, as the name on the Taylor series centred at zero. That series was not his discovery, and he never claimed it. His real achievements were larger: the most careful account of the calculus written in eighteenth-century Britain, the integral test for series, one half of the Euler–Maclaurin summation formula, the theory of the attraction of ellipsoids, and the first exact equilibrium figures of a rotating self-gravitating fluid. Joseph-Louis Lagrange admired his geometry of attractions without reservation.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proofs or derivations, and exercises with folded solutions. The order follows his career: geometry of curves (1720), algebra (1726–1729), the Treatise of Fluxions (1742), and the gravitational problems that ran through his work from the 1740 tides essay onwards.

Timeline

Argyll and Glasgow, 1698–1717

Colin Maclaurin was born in February 1698 at Kilmodan in Glendaruel, in the Cowal district of Argyll, the youngest of three sons of John Maclaurin, the parish minister. His father died when Colin was about six weeks old, and his mother died in 1707. Their uncle Daniel Maclaurin, minister of Kilfinan on Loch Fyne, then brought up Colin and his brother John.

In 1709, aged eleven, Colin entered the University of Glasgow, intending like his father and brother to enter the ministry. Scottish students then began young, and Glasgow had a strong tradition in Latin and philosophy. There he met Robert Simson, ten years older, who was turning to the restoration of Greek geometry and later became professor of mathematics at Glasgow. Simson’s love of synthetic geometry stayed with Maclaurin all his life. Maclaurin graduated MA at about fourteen, defending in public a Latin thesis on the power of gravity, a remarkably Newtonian topic for a Scottish student at that date.

After graduating he spent several years at his uncle’s manse in Kilfinan, studying divinity and mathematics. When he applied for a chair in 1717 he was still described as a student of divinity.

Aberdeen, London and the Continent, 1717–1725

In August 1717 Marischal College, Aberdeen, held a competitive trial lasting ten days for its vacant chair of mathematics. Maclaurin won it at nineteen. He held the record as the world’s youngest professor until 2008.

Newton

In 1719 he went to London with the manuscript of his first book. He met Isaac Newton, then President of the Royal Society, who took an interest in him that lasted the rest of Newton’s life. He was elected a Fellow of the Royal Society that year, with Edmond Halley among his proposers. Geometria Organica; sive descriptio linearum curvarum universalis appeared in London in 1720 (Lesson 04).

Abroad

In 1722 he accepted a post as travelling tutor to the son of Lord Polwarth, without first arranging cover for his classes or waiting for his college’s consent. They spent time in France. In this period Maclaurin also wrote an essay on the percussion of bodies that won a prize of the Paris Academy of Sciences. The tour ended when his pupil died at Montpellier, and Maclaurin returned to Aberdeen in 1724. The college called him to account for his absence. He was allowed to resume, but his relations with the authorities in Aberdeen stayed difficult.

In 1725 the Edinburgh chair, held by the elderly James Gregory, nephew of the inventor of the Gregory series, needed a deputy. Newton wrote in support of Maclaurin in August, and in November wrote to the Lord Provost offering to contribute £20 a year towards his salary. There is no evidence the offer was taken up. Maclaurin began teaching in Edinburgh on 3 November 1725 as joint professor with Gregory, whom he soon succeeded.

Edinburgh, 1725–1745

Maclaurin spent the rest of his career at Edinburgh, where he built up the teaching of mathematics. He taught geometry, algebra, fluxions, astronomy and natural philosophy, in the Newtonian spirit. He married Anne Stewart, daughter of Walter Stewart, the Solicitor General for Scotland, on 8 July 1733. They had seven children, of whom two sons and three daughters survived him. Their eldest son John became a judge with the title Lord Dreghorn and was one of the founders of the Royal Society of Edinburgh.

Public mathematics

Maclaurin applied mathematics wherever he was asked. He helped customs officers at Glasgow calculate the contents of molasses casks for duty. In 1737 he led the expansion of an Edinburgh medical society into the Philosophical Society of Edinburgh, a general scientific society. After a long gap caused by the troubles of 1745 it became, under royal charter in 1783, the Royal Society of Edinburgh.

Quarrels

He was a generous teacher and a sensitive author. He disputed priority with George Campbell over results on imaginary roots of equations, and accused William Braikenridge of publishing as his own theorems on the description of curves that Maclaurin had found earlier (Lesson 04). His friendship with James Stirling, by contrast, was warm, and the two exchanged ideas on series for years.

The Treatise

In 1734 George Berkeley’s The Analyst attacked the logic of fluxions. Maclaurin set out to answer it with a complete treatise built on the methods of Archimedes. The work grew for eight years. When A Treatise of Fluxions finally appeared in Edinburgh in 1742, its two volumes ran to 763 pages and contained far more than a defence: series, maxima and minima, the attraction of ellipsoids, the figure of the Earth and the tides.

Organic description of curves

For the general reader. A compass draws a circle because it rotates a fixed length about a point. Newton showed that rotating two fixed angles about two pins, with the crossing point of one pair of arms kept on a straight line, makes the other pair of arms trace a conic. Maclaurin’s first book extended this mechanical, or organic, description to curves of higher degree. Along the way he noticed a paradox about how many points determine a curve.

Counting points

A plane curve of degree \(n\) is the zero set of a polynomial \(F(x,y)=\sum_{i+j\le n}a_{ij}x^iy^j\). There are \(\binom{n+2}2\) coefficients, and multiplying \(F\) by a non-zero constant gives the same curve, so curves of degree \(n\) form a projective space of dimension \(N=\frac{n(n+3)}2\). Each condition “passes through a given point” is linear in the coefficients, so in general \(N\) points determine a unique curve of degree \(n\). Bézout’s theorem says two curves of degrees \(m,n\) without common component meet in at most \(mn\) points.

Gabriel Cramer discussed the puzzle in 1750 and credited Maclaurin, but the name Cramer’s paradox stuck. Leonhard Euler explained it in correspondence with Cramer in the 1740s. The resolution is that the \(n^2\) intersection points of two curves are never in general position: the linear conditions they impose are dependent. For cubics, any cubic through 8 of the 9 intersection points of two cubics passes through the ninth (the Cayley–Bacharach theorem), so the 9 points impose only 8 conditions and determine a pencil \(\lambda F+\mu G\) of cubics.

The Braikenridge–Maclaurin theorem

The theorem is the converse of Blaise Pascal’s hexagon theorem, and it yields a ruler-only construction of the conic through five given points. Both Maclaurin and Braikenridge published versions in the 1730s, and their priority dispute was bitter.

Fluxions without infinitesimals

For the general reader. Berkeley had mocked the calculus for dividing by quantities and then treating them as zero. Maclaurin’s answer was to rebuild it on the methods of Archimedes, using only finite quantities, velocities and proofs by contradiction. His treatise also gave the first clear test for maxima and minima using higher derivatives, and the series that now carries his name.

The method

Maclaurin took a fluxion to be the velocity with which a quantity is generated by motion. His basic axioms compared quantities generated at greater or lesser velocities over equal times. To prove that a fluxion has a given value, he showed that it can be neither greater nor less, each alternative leading to contradiction, as Archimedes had done for areas. The method is long but sound. It anticipates the use of inequalities, rather than vanishing quantities, that Cauchy and Weierstrass later turned into the definition of the limit.

The series

In Book II Maclaurin assumes that a function \(f\) can be written as \(f(x)=A+Bx+Cx^2+Dx^3+\cdots\), differentiates term by term, and sets \(x=0\) to find \(A=f(0)\), \(B=f'(0)\), \(C=f''(0)/2\), and so on. He credited the result to Brook Taylor (1715) and to James Stirling. The modern statement with remainder is:

Maclaurin did not discuss convergence. The function \(f(x)=e^{-1/x^2}\) (with \(f(0)=0\)) has all derivatives zero at 0, so its Maclaurin series is identically zero, although \(f\) is not. Augustin-Louis Cauchy gave this example in 1823.

Maxima and minima

Proof. By Taylor’s theorem, \(f(a+h)-f(a)=\frac{f^{(k)}(\xi)}{k!}h^k\) with \(\xi\) between \(a\) and \(a+h\). For small \(h\), \(f^{(k)}(\xi)\) has the sign of \(f^{(k)}(a)\). If \(k\) is even, \(h^k>0\) for \(h\neq0\); if \(k\) is odd, \(h^k\) changes sign with \(h\). \(\square\)

Series, sums and integrals

For the general reader. A sum of many terms and the area under a smooth curve are close relatives. Maclaurin turned this into two tools. The integral test decides whether an infinite series converges by comparing it with an area. The summation formula he found independently of Euler gives the exact difference between sum and area as a series of correction terms, and computes slowly converging sums with startling accuracy.

Proof. Since \(f\) decreases, \(f(k+1)\le\int_k^{k+1}f\le f(k)\). Sum over \(k\). \(\square\)

The summation formula

Let \(B_k\) be the Bernoulli numbers, defined by \(\frac{t}{e^t-1}=\sum_k B_k\frac{t^k}{k!}\), so \(B_1=-\tfrac12,\ B_2=\tfrac16,\ B_4=-\tfrac1{30},\ B_6=\tfrac1{42}\), and \(B_{2j+1}=0\) for \(j\ge1\).

Sketch. On \([k,k+1]\), integrate by parts against \(B_1(x-k)=x-k-\tfrac12\): \(\int_k^{k+1}f=\frac{f(k)+f(k+1)}2-\int_k^{k+1}B_1(x-k)f'(x)\,dx\). Summing gives the trapezoidal rule plus an error integral. Integrate the error by parts repeatedly, using \(B_{r+1}'=(r+1)B_r\) and \(B_r(0)=B_r(1)\) for \(r\ge2\). Each step produces one boundary term. \(\square\)

Euler found the formula around 1732–1735 and used it to sum series. Maclaurin derived it independently in the Treatise of Fluxions, mainly as a tool to evaluate integrals from sums. Like Stirling’s series, the full expansion usually diverges; it is asymptotic, and one stops near the smallest term.

Roots and symmetric means

For the general reader. The coefficients of a polynomial are built symmetrically from its roots. Newton noticed that when all the roots are real, the coefficients must satisfy certain inequalities, which gives a quick way to detect complex roots. Maclaurin wrote two long letters on this theory in 1729, and the chain of inequalities between averages of products of numbers now bears his name. His posthumous Treatise of Algebra also contains the rule for solving linear systems later named after Cramer.

For positive reals \(x_1,\dots,x_n\) let \(e_k\) be the \(k\)-th elementary symmetric polynomial and \(S_k=e_k/\binom nk\) its average, with \(S_0=1\). Then \(S_1\) is the arithmetic mean and \(S_n^{1/n}\) the geometric mean.

Proof from Newton’s inequalities. All \(S_k>0\). Raise \(S_{j-1}S_{j+1}\le S_j^2\) to the power \(j\) and multiply for \(j=1,\dots,k\). On the left, \(S_i\) for \(1\le i\le k-1\) appears with exponent \((i+1)+(i-1)=2i\), \(S_k\) with exponent \(k-1\), and \(S_{k+1}\) with exponent \(k\). On the right, \(S_i\) appears with exponent \(2i\). Cancelling leaves \(S_k^{k-1}S_{k+1}^{k}\le S_k^{2k}\), so \(S_{k+1}^{k}\le S_k^{k+1}\), that is \(S_{k+1}^{1/(k+1)}\le S_k^{1/k}\). \(\square\)

The first link, \(S_1\ge S_n^{1/n}\), is the inequality of arithmetic and geometric means.

Repeated roots and the derived equation

Maclaurin’s 1729 letters use the derived polynomial \(f'\) to locate equal roots: \(r\) is a repeated root of \(f\) exactly when \(f(r)=f'(r)=0\), so the repeated roots are the roots of \(\gcd(f,f')\). Between two real roots of \(f\) there is a real root of \(f'\), which bounds the number of real roots of \(f\) by one more than the number for \(f'\).

Linear systems

The Treatise of Algebra (1748) gives, for two and three unknowns, the solution of a linear system as ratios of sums of signed products of coefficients, the determinants of today. For two unknowns, \(ax+by=c,\ dx+ey=f\) gives

Gabriel Cramer stated the general rule for \(n\) unknowns in 1750.

Attraction of ellipsoids

For the general reader. Newton proved that a uniform sphere attracts outside points as if all its mass were at its centre. The Earth is not a sphere but a slightly flattened spheroid, and to know its shape one must know how such a body attracts. Maclaurin solved the problem for points inside a uniform ellipsoid, using pure geometry, and proved a striking theorem for points outside. Lagrange called this part of his work a masterpiece of geometry.

The homoeoid

Proof. Through an interior point \(P\), draw a thin double cone of small solid angle \(d\Omega\). It cuts the shell in two pieces, at distances \(r_1\) and \(r_2\) from \(P\). Their masses are \(\rho\, r_i^2\,d\Omega\,t_i\), where \(t_i\) is the thickness along the chord, and they attract \(P\) with forces \(G\rho\,t_i\,d\Omega\) in opposite directions. Any chord of the inner ellipsoid and the outer, similar ellipsoid have the same midpoint (both midpoints lie on the diametral plane conjugate to the chord direction). So the two intercepts along the chord are equal, \(t_1=t_2\), and the forces cancel. \(\square\)

Inside a solid ellipsoid

Let the ellipsoid be \(Q(\mathbf r)=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\le1\), with density \(\rho\), and let \(B(\mathbf r,\mathbf u)\) be the symmetric bilinear form with \(B(\mathbf r,\mathbf r)=Q(\mathbf r)\). The same double-cone argument works at any interior point \(P\). The line \(P+t\mathbf u\) leaves the ellipsoid where \(Q(\mathbf u)t^2+2B(P,\mathbf u)t+Q(P)-1=0\). The two cones of solid angle \(d\Omega\) along \(\pm\mathbf u\) attract \(P\) with net force \(G\rho\,d\Omega\,(t_++t_-)\,\mathbf u\), where \(t_\pm\) are the roots. Their sum, \(-2B(P,\mathbf u)/Q(\mathbf u)\), is linear in \(P\). Integrating over directions shows that the attraction is a linear function of position. By the symmetry of the ellipsoid in its three coordinate planes it is diagonal:

where the constants \(A_i\) depend only on the axis ratios. For a sphere \(A_1=A_2=A_3=\tfrac43\), recovering \(g=-\tfrac43\pi G\rho\,r\). For an oblate spheroid \(a=b>c\) with eccentricity \(e\) (\(c^2=a^2(1-e^2)\)), the standard values are

with \(2A_1+A_3=4\), the trace identity that follows from Poisson’s equation \(\nabla\cdot\mathbf g=-4\pi G\rho\).

Laplace and Adrien-Marie Legendre extended the result to all external points in the 1780s, and James Ivory gave a simpler proof in 1809. The theorem reduces the external attraction of any homogeneous ellipsoid to the internal one, which Maclaurin had solved.

Rotating fluids and the figure of the Earth

For the general reader. Newton argued that the spinning Earth must bulge at the equator, and estimated the flattening at about 1 part in 230. His argument assumed, without proof, that the shape would be a spheroid. Maclaurin proved that a uniform fluid spheroid rotating steadily is an exact equilibrium, and found the relation between its spin and its flattening. These shapes are now called Maclaurin spheroids. The question of what other shapes are possible led to Jacobi, Poincaré and the theory of bifurcation.

Equilibrium

In the frame rotating with angular velocity \(\Omega\) about the \(z\)-axis, a fluid at rest is in equilibrium if its surface is a level surface of the total potential, gravitational plus centrifugal. Using Lesson 08, the gravitational potential inside is \(\Phi=\pi G\rho\,\big(A_1(x^2+y^2)+A_3z^2\big)/2+\text{const}\), and the centrifugal potential is \(-\tfrac12\Omega^2(x^2+y^2)\). The total is constant on the surface \(\frac{x^2+y^2}{a^2}+\frac{z^2}{c^2}=1\) exactly when the coefficients are in the ratio \(1/a^2:1/c^2\):

The right side rises from 0 at \(e=0\) to a maximum of about \(0.4493\) at \(e\approx0.9300\), then falls back to 0 as \(e\to1\). So a uniform fluid cannot rotate faster than \(\Omega^2=0.4493\,\pi G\rho\), and each slower rate allows two spheroids, one slightly and one strongly flattened.

Later history

In 1834 Carl Gustav Jacob Jacobi showed that rotating three-axis ellipsoids can also be in equilibrium. They branch off the Maclaurin sequence at \(e\approx0.8127\), where \(\Omega^2/\pi G\rho\approx0.3742\); beyond that point a slightly viscous Maclaurin spheroid becomes secularly unstable. Henri Poincaré followed the sequence further in 1885, to pear-shaped figures. The modern account is Subrahmanyan Chandrasekhar’s Ellipsoidal Figures of Equilibrium (1969).

The tides

For the general reader. For 1740 the Paris Academy set the tides as its prize question. It shared the prize among Daniel Bernoulli, Leonhard Euler and Maclaurin, whose essays built on Newton’s gravity, and a fourth, Cartesian entrant. Maclaurin showed that a deep fluid layer on a smooth sphere, pulled by a distant body, takes the form of an elongated spheroid pointing at the body. He used the same theory of ellipsoids as in Lesson 09.

The tidal potential

A body of mass \(M\) at distance \(d\) produces at a point at distance \(r\ll d\) from the Earth’s centre, angle \(\vartheta\) from the line to the body, the potential \(-GM/|\mathbf d-\mathbf r|\). Expanding in \(r/d\) and discarding the constant and the uniform part that accelerates the Earth as a whole leaves

This is a quadratic function of position, with the same form as the centrifugal potential of Lesson 09 but symmetric about the line to the body. The equilibrium surface is therefore again an ellipsoid, here prolate, pointed towards and away from the body.

Sketch. On a rigid sphere the surface rises by \(-W/g\) with \(g=GM_E/R^2\); the difference between \(\vartheta=0\) and \(\vartheta=\pi/2\) is \(\frac{GMR^2}{d^3g}\cdot\frac32=h_0\). For the fluid Earth the bulge adds its own potential, and Maclaurin’s formulas for the attraction of a nearly spherical ellipsoid give the factor \(\frac52\), the same factor that turns Huygens’s rotational flattening \(\frac12m\) into Newton’s \(\frac54m\). \(\square\)

Widows, rebels and the last year, 1743–1746

The widows’ fund

In the early 1740s two Church of Scotland ministers, Robert Wallace and Alexander Webster, designed a fund to support the widows and orphans of ministers and of university professors. It was to be a funded scheme, with contributions fixed in advance from estimates of how many ministers would die each year, how many widows they would leave, and for how long the widows would draw pensions. Maclaurin checked and refined their calculations, corresponded with Wallace about the scheme in 1743, and went to London to support the bill. The General Assembly approved the plan in 1743, and an Act of Parliament (17 Geo. II c. 11) passed in the session of 1743–44 established it. The episode is often cited as an early example of actuarial science.

The ’45

In 1745 Charles Edward Stuart landed in Scotland, and the Jacobite army marched on Edinburgh. The city walls were weak. Maclaurin, an ardent supporter of the Hanoverian government, threw himself into planning and building defence works, working long hours in the trenches. The city fell to the Jacobites in September without serious resistance. Maclaurin fled to England, where the Archbishop of York invited him to stay. He returned to Edinburgh in November after the Jacobite army left for England.

The exposure and exertion of the autumn broke his health. He died in Edinburgh on 14 June 1746, aged 48, two months after the Jacobites’ defeat at Culloden. He is buried in Greyfriars Kirkyard, where a table stone records only his initials and the years of birth and death.

Legacy

Two books appeared posthumously in 1748. A Treatise of Algebra, based on his lectures and modelled on Newton’s Arithmetica Universalis, became a standard text and included the rule for linear systems discussed in Lesson 07. An Account of Sir Isaac Newton’s Philosophical Discoveries, edited by Patrick Murdoch with a memoir of the author and printed for Maclaurin’s children, gave a clear exposition of Newtonian natural philosophy for educated readers and was translated into French in 1749.

Maclaurin’s geometrical style shaped British mathematics for two generations. Its rigour was admired, but its avoidance of Leibniz’s notation, followed by most British mathematicians, has been blamed for the isolation of British analysis until the reforms of the Analytical Society in Cambridge in the 1810s. On the Continent, his attraction theorems and rotating spheroids were taken up by Alexis Clairaut, Lagrange, Pierre-Simon Laplace and later Jacobi and Poincaré.

• Analysis: the integral test; the Euler–Maclaurin formula; the higher-derivative test for extrema; a rigorous treatment of fluxions without infinitesimals.
• Algebra and geometry: Maclaurin’s inequality; early determinant solutions of linear systems; the Braikenridge–Maclaurin theorem; the paradox of points and curves.
• Physics: the attraction of ellipsoids, Maclaurin spheroids, and the equilibrium theory of tides.
• Institutions: the Philosophical Society of Edinburgh, forerunner of the Royal Society of Edinburgh; the actuarial basis of the Scottish ministers’ widows’ fund.
• Memorials: the Maclaurin Society at Glasgow, Colin Maclaurin Road at the University of Edinburgh’s King’s Buildings, and a Heriot-Watt University building bear his name.

Connections

People linked to Colin Maclaurin across this collection. A name without a link has no page here yet. See Colin Maclaurin in the network of all 337 lives →

Influenced by

• Brook Taylor
• Isaac Newton 1719-1727
• Robert Simson 1709-1713

Influenced

• Aleksandr Lyapunov
• Alexis Clairaut
• Joseph-Louis Lagrange
• Gabriel Cramer

Colleagues and collaborators

• Daniel Bernoulli 1740
• Edmond Halley
• Leonhard Euler 1740

Rivals

• George Berkeley
• William Braikenridge

Friends

• James Stirling
• Patrick Murdoch

Also mentioned in this life

• Archimedes
• Augustin-Louis Cauchy
• Arthur Cayley
• Subrahmanyan Chandrasekhar
• Christiaan Huygens
• Carl Gustav Jacob Jacobi
• Pierre-Simon Laplace
• Adrien-Marie Legendre
• Gottfried Wilhelm Leibniz
• Max Noether
• Blaise Pascal
• Henri Poincaré
• Siméon Denis Poisson
• Karl Weierstrass

Fields

• calculus 9 lives
• analysis 20 lives
• geometry 32 lives
• algebra 29 lives
• mathematical physics 45 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Colin Maclaurin, MacTutor History of Mathematics.
2. Colin Maclaurin, Wikipedia.
3. The prickly genius: Colin MacLaurin (1698–1746), Mathematical Gazette.
4. W. W. Rouse Ball, Colin Maclaurin, from A Short Account of the History of Mathematics (1908).
5. P. Murdoch, An account of the life and writings of the author, prefixed to Maclaurin’s Account of Sir Isaac Newton’s Philosophical Discoveries, London, 1748.
6. S. Mills and others (eds.), The Collected Letters of Colin MacLaurin, Shiva, 1982.
7. J. V. Grabiner, Was Newton’s calculus a dead end? The continental influence of Maclaurin’s Treatise of Fluxions, American Mathematical Monthly 104 (1997) 393–410.

Mathematics

1. C. Maclaurin, Geometria Organica, London, 1720.
2. C. Maclaurin, A second letter to Martin Folkes, Esq., concerning the roots of equations, Phil. Trans. R. Soc. 36 (1729) 59–96.
3. C. Maclaurin, A Treatise of Fluxions, 2 vols., Edinburgh, 1742.
4. C. Maclaurin, A Treatise of Algebra, London, 1748.
5. Euler–Maclaurin formula; Maclaurin’s inequality; Cramer’s paradox; Maclaurin spheroid, Wikipedia.
6. S. Chandrasekhar, Ellipsoidal Figures of Equilibrium, Yale, 1969.
7. I. Todhunter, A History of the Mathematical Theories of Attraction and the Figure of the Earth, 1873.