Mary Somerville

How to use this course

Mary Somerville made no great discovery, and she did not claim one. Her achievement was of another kind. She mastered the hardest mathematical physics of her time, Pierre-Simon Laplace’s Traité de mécanique céleste, without a teacher or a university, and explained it in English to a generation of readers. Her four books ran to many editions. One of them gave the occasion for the word “scientist”, and another pointed to the possibility of a planet beyond Uranus a few years before Neptune was found.

Part II follows the content of her books, especially The Mechanism of the Heavens (1831) and On the Connexion of the Physical Sciences (1834). It teaches the physics she explained: the gravitational potential and Laplace’s equation, the shape of the Earth, the tides, the long-term stability of planetary orbits, the perturbations of Uranus, the invisible parts of the solar spectrum and the physics of the atmosphere.

Each technical lesson opens with an italic paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Companion modules cover her friend Caroline Herschel, who was elected to the Royal Astronomical Society with her, and William Herschel, whose discovery of infrared light figures in Lesson 09.

Timeline

Burntisland, 1780–1797

Mary Fairfax was born on 26 December 1780 at the manse in Jedburgh, the home of her mother’s sister and her husband, the Reverend Thomas Somerville. Her father, William George Fairfax, was a naval officer often at sea; he was knighted after the battle of Camperdown in 1797 and rose to vice-admiral. Her mother, Margaret Charters, brought up the children in Burntisland, a small port on the Fife shore of the Firth of Forth. Mary ran free on the beach, collecting shells and watching the birds and the tides.

When her father returned from a voyage and found that, at about ten, she could barely read, he sent her to a boarding school in Musselburgh for a year. She hated it and learned little beyond reading. At home again she read everything she could find, taught herself enough Latin to read Caesar, and in Jedburgh studied with her uncle Thomas Somerville, who encouraged her.

Euclid by accident

At about thirteen she came upon algebraic symbols in a puzzle page of a ladies’ fashion magazine, and wanted to know what they meant. Soon after, in Edinburgh, she attended the painting classes of Alexander Nasmyth. She heard him tell another pupil that Euclid’s Elements were the foundation of perspective, of astronomy and of all mechanical science. She persuaded her younger brother’s tutor to buy her a copy of Euclid and an algebra book. Her family disapproved, and the servants were told to take away her candles. She worked through the first six books of Euclid at night, from memory, in the dark.

Two marriages and a library, 1797–1816

In 1804 Mary Fairfax married her cousin Samuel Greig, a captain in the Russian navy posted to London, and went to live there. Greig had a low opinion of learned women and gave her no encouragement. They had two sons, born in 1805 and 1806. Greig died in 1807, and she returned to Scotland as a young widow with an independent income and, for the first time, freedom to study as she pleased.

In Edinburgh she found mentors. John Playfair, professor of natural philosophy at the university, encouraged her. William Wallace, then a master at the Royal Military College and later professor of mathematics at Edinburgh, corresponded with her and recommended a course of reading: the calculus and mechanics of the French school, culminating in Laplace’s Mécanique céleste. She bought the books and worked through them alone. She solved problems posed in the Mathematical Repository, a journal edited by Thomas Leybourn, and in 1811 won a silver medal for one of her solutions.

In 1812 she married another cousin, William Somerville, son of the uncle who had taught her Latin. He was an army surgeon and an inspector of hospitals, and, unlike Greig, he was proud of her mathematics and helped her work. They had four children: Margaret (1813–1823), Thomas (1814–1815), Martha (1815–1879) and Mary Charlotte (1817–1875). Of her six children, three survived to adulthood: Woronzow Greig, Martha and Mary.

London, Paris and Laplace, 1816–1831

In 1816 William Somerville joined the Army Medical Board and the family moved to London. They soon belonged to the city’s scientific society. Their friends included William and John Herschel, George Airy, George Peacock and Charles Babbage, whose calculating engines Mary visited. In 1817 they toured the Continent. In Paris, with introductions from Jean-Baptiste Biot and François Arago, she met Laplace, Siméon Denis Poisson and Louis Poinsot, and she kept up a correspondence with several of them afterwards. In 1824 William became physician to the Royal Hospital at Chelsea, and they lived there.

The first paper and the commission

In 1825 she made experiments to test whether violet sunlight could magnetize a steel needle, as some Italian physicists claimed. She reported a positive result, and in 1826 the Royal Society published her paper, presented by her husband. The effect was later shown to be spurious (Lesson 09).

In March 1827 Henry Brougham, founder of the Society for the Diffusion of Useful Knowledge, wrote to William asking whether Mary would prepare an English account of the Mécanique céleste for the society’s cheap educational series. She agreed, on condition that the manuscript be burned if it failed. She worked between household duties and visitors. The book outgrew the series. On John Herschel’s advice it was published separately by John Murray in 1831 as The Mechanism of the Heavens. It did not simply translate Laplace. It filled in the steps he had omitted, often with the comment that it was easy to see, and added diagrams and a long preliminary dissertation for the general reader, published separately in 1832. The book made her famous. David Brewster called her “certainly the most extraordinary woman in Europe.”

The attraction of bodies and Laplace’s equation

For the general reader. Newton showed that a planet attracts as if all its mass were at its centre, provided it is a perfect sphere. Real planets are flattened, lumpy and spinning. Laplace’s great tool for such bodies was a single equation satisfied by the gravitational potential in empty space. Somerville’s Mechanism opened Laplace’s theory to English readers through this equation and the special functions, now called spherical harmonics, that solve it.

Sketch. A direct computation in spherical coordinates shows \(\nabla^2(1/r)=\frac1{r^2}\frac{d}{dr}\big(r^2\cdot(-1/r^2)\big)=0\) for \(r\ne0\), so each term of the integral is harmonic away from its source. Outside the masses one may differentiate under the integral. Inside, the flux of \(\nabla(1/r)\) through any small sphere about the origin is \(-4\pi\), which gives \(\nabla^2(1/r)=-4\pi\delta\) and hence Poisson’s equation. Laplace stated only the outside equation; Poisson added the right-hand side in 1813. \(\square\)

Proof via Laplace’s equation. Outside the body, \(V\) depends only on \(r\) and is harmonic, so \(\frac{1}{r^2}(r^2V')'=0\), giving \(V=A+B/r\). As \(r\to\infty\), \(V\to0\), so \(A=0\). By Gauss’s theorem, the flux of \(\nabla V\) through a sphere enclosing the body is \(4\pi GM\), so \(B=-GM\). \(\square\)

Legendre expansion

For a body symmetric about the \(z\)-axis, the general solution of Laplace’s equation outside it, decaying at infinity, is a series in Legendre polynomials \(P_n\):

Here \(R\) is the equatorial radius, \(\theta\) the colatitude, and the \(n=1\) term vanishes when the origin is at the centre of mass. The dimensionless coefficient \(J_2\) measures the equatorial bulge. For the Earth \(J_2\approx1.083\times10^{-3}\).

The figure of the Earth

For the general reader. A spinning fluid planet bulges at the equator. Newton predicted the Earth’s flattening, and French expeditions to Lapland and Peru in the 1730s confirmed it. Alexis Clairaut then found a formula that links the flattening to the way gravity changes from equator to pole, so that pendulum measurements could weigh in on the shape of the planet. Laplace extended the theory, and Somerville’s Mechanism devoted a book to it.

Let the Earth have equatorial radius \(a\), polar radius \(c\), and flattening \(f=(a-c)/a\). Let \(\omega\) be its angular velocity and \(g_e\) gravity at the equator. The ratio of centrifugal acceleration to gravity at the equator is

Sketch. Outside the body the effective potential is \(U=-\frac{GM}{r}+\frac{GMJ_2a^2}{r^3}P_2(\sin\varphi)-\frac12\omega^2r^2\cos^2\varphi\), and to first order \(m=\omega^2a^3/GM\). The surface is an equipotential. Write \(r=a(1-f\sin^2\varphi)\) and require \(U\) to be the same at equator and pole: this gives \(f=\tfrac32J_2+\tfrac12m\), i.e. \(J_2=\tfrac13(2f-m)\). The magnitude of gravity is \(\partial U/\partial r\). At the equator, \(g_e=\frac{GM}{a^2}\big(1+\tfrac32J_2-m\big)\). At the pole, \(r=a(1-f)\) and \(g_p=\frac{GM}{a^2}\big(1+2f-3J_2\big)\). Hence

Only the external field enters, so the result does not depend on how density varies with depth, which is what makes it powerful. \(\square\)

For a uniform incompressible fluid, Newton found \(f=\tfrac54m\approx1/230\). The real Earth is more centrally condensed, and \(f\approx1/298\).

The tides

For the general reader. The Moon pulls harder on the near side of the Earth than on the centre, and harder on the centre than on the far side. The result is a stretching force that raises two bulges of water, one facing the Moon and one opposite, and the Earth turns beneath them. The Sun does the same, less strongly. Newton explained this, and Laplace turned it into a dynamical theory of the oceans. Somerville explained both in the Mechanism and the Connexion, and she knew the tides of the Forth from childhood.

Proof. Expand \(-GM/|\mathbf d-\mathbf r|=-\frac{GM}{d}\sum_n(r/d)^nP_n(\cos\psi)\). The \(n=0\) term is constant. The \(n=1\) term, \(-GMr\cos\psi/d^2\), gives a uniform force equal to the one that accelerates the whole Earth, and so has no effect relative to the Earth. The \(n=2\) term remains. \(\square\)

The surface rises by \(h\) where \(g h=-W\), and \(g=GM_E/R^2\). The real tides differ greatly from this equilibrium value, because the oceans have a natural response time comparable to the forcing and are broken up by continents. Laplace’s tidal equations of 1775–76 treat the ocean as a thin rotating fluid layer forced by \(W\). They are the starting point of modern tidal theory.

Jupiter, Saturn and the stability of the system

For the general reader. Comparing ancient and modern observations, astronomers had found that Jupiter seemed to be speeding up and Saturn slowing down. If this went on, Jupiter would fall towards the Sun and Saturn would drift away. Newton had suspected that the solar system might need divine correction. Laplace showed in 1785 that the effect is a long oscillation of about 900 years, caused by the near ratio of 5 to 2 between their periods. The result was a showpiece of the Mécanique céleste, and Somerville explained it with care.

Sketch. By Lagrange’s planetary equations, \(\dot a=\frac{2}{na}\frac{\partial\mathcal R}{\partial\lambda}\), where \(\mathcal R\) is the disturbing function and \(\lambda\) the mean longitude. Expand \(\mathcal R\) as a Fourier series in the mean longitudes, \(\mathcal R=\sum C_{j,k}\cos(j\lambda+k\lambda'+\ldots)\). The terms independent of \(\lambda\) and \(\lambda'\) (the secular part) have zero derivative with respect to \(\lambda\). So every term in \(\dot a\) oscillates with a frequency \(jn+kn'\ne0\). \(\square\)

Small divisors

Integrating a term \(\cos\big((jn+kn')t\big)\) brings in the factor \(1/(jn+kn')\) in \(a\), and in the longitude, which is obtained by integrating \(n\), the factor \(1/(jn+kn')^2\). When \(jn+kn'\) is nearly zero, a small coefficient can produce a large, slow inequality. For Jupiter and Saturn, \(2n_J-5n_S\) is small.

Because the interaction energy is periodic and small, the sum of the two orbital energies, \(-\frac{GM_\odot m_J}{2a_J}-\frac{GM_\odot m_S}{2a_S}\), is nearly constant. When Jupiter’s orbit shrinks (and its mean motion grows), Saturn’s must expand (and its mean motion fall).

Uranus and the unseen planet

For the general reader. After William Herschel found Uranus in 1781, astronomers computed its orbit, allowing for the pulls of Jupiter and Saturn. By the 1830s the planet was clearly straying from the tables. In the sixth edition of the Connexion (1842), Somerville wrote that the discrepancies might one day reveal the existence and even the mass and orbit of a planet beyond the reach of sight. In 1845–46 John Couch Adams and Urbain Le Verrier independently computed where such a planet should be, and on 23 September 1846 Johann Galle found Neptune within about a degree of Le Verrier’s position.

The relative angle grows at \(2\pi(1/P_1-1/P_2)\) per unit time, which gives the formula. For Uranus and Neptune it is long, so the interval 1781–1846 covered only one conjunction, around 1822. Before it, the faster Uranus was catching up with Neptune, which lay ahead and pulled it forward along its orbit; after it, Neptune lay behind and held it back. This is the pattern in the residuals that Adams and Le Verrier explained, shown in the figure at the top of this page.

Neither Adams nor Le Verrier knew Neptune’s distance. Both started from the Titius–Bode rule, \(a_n=0.4+0.3\times2^n\) AU, which had fitted Uranus well. For the next planet it gave 38.8 AU, while Neptune’s true distance is 30.1 AU. Their predicted orbits were too large, but they placed the planet correctly in the 1840s, because the direction of the pull near conjunction depends mostly on where the planet is, and less on how far.

Rays beyond the visible

For the general reader. In 1800 William Herschel found invisible heat rays beyond the red end of the Sun’s spectrum, and in 1801 Johann Wilhelm Ritter found invisible rays beyond the violet that darken silver salts. Light, heat and chemical action seemed to be different agents, all coming from the Sun. Somerville’s first experiment asked whether violet light could also magnetize steel. She thought it could; later workers could not reproduce her result. A real link between light and magnetism was found by Michael Faraday in 1845. This lesson explains why the ends of the spectrum act so differently.

Her experiment

Following claims by Domenico Morichini in Rome, she exposed half of a steel sewing needle to violet light from a prism while covering the other half, and then tested it for magnetism with a compass. The needle appeared magnetized, and her paper of 1826 reported the result. Within a few years other experimenters, among them Peter Riess and Ludwig Moser, repeated such experiments with careful controls and found no effect. Magnetism picked up from the surroundings, or from handling, probably explains the positive results. She later studied the action of different parts of the spectrum on paper treated with plant juices, a more fruitful question that connects with early photography.

Derivation of Wien’s law. Planck’s law per unit wavelength is \(B_\lambda\propto\lambda^{-5}(e^{x}-1)^{-1}\) with \(x=hc/\lambda kT\). Setting \(dB_\lambda/d\lambda=0\) gives \(5(1-e^{-x})=x\), whose positive root is \(x\approx4.965\). So \(\lambda_{\max}T=hc/(4.965k)\). \(\square\)

Physical geography: the atmosphere

For the general reader. Physical Geography (1848) described the Earth as a single system of land, sea, air and life. It was the first English textbook on the subject and was used in schools and universities for decades. Two simple physical laws govern much of what it says about the air: pressure falls by a constant factor for each fixed step upward, and rising air cools as it expands. Together they explain why mountains are cold and why snow lies on the peaks of the tropics.

Proof. Hydrostatic equilibrium gives \(dp/dz=-\rho g\), and the ideal gas law gives \(\rho=p\mu/(RT)\). So \(dp/p=-dz/H\). \(\square\)

Proof. For an adiabatic change of an ideal gas, the first law gives \(c_p\,dT=dp/\rho\). With \(dp=-\rho g\,dz\) from hydrostatic balance, \(c_p\,dT=-g\,dz\). \(\square\)

Moist air cools more slowly, because condensing water releases latent heat. The observed average fall in the lower atmosphere is about 6.5 K per kilometre. In the nineteenth century barometers were the standard way to measure the heights of mountains, using a version of the first proposition with corrections for temperature.

The Connexion and fame, 1831–1838

In 1834 Somerville published On the Connexion of the Physical Sciences. It had no equations. It presented astronomy, light, heat, electricity, magnetism and sound as aspects of a few general principles, and argued that the sciences were growing together. The book went through ten editions, and she revised it for each, bringing in the latest work on electromagnetism, light and astronomy.

William Whewell reviewed it in the Quarterly Review in 1834. Discussing the lack of a general name for those who studied nature, he reported a suggestion of “scientist”, by analogy with “artist”. It is the first known use of the word in English. Whewell used the occasion of Somerville’s book to argue that the sciences needed people who saw them whole.

Recognition

Honours followed quickly. In 1834 she was elected an honorary member of the Royal Irish Academy and of the Société de Physique et d’Histoire Naturelle of Geneva. In 1835 the Royal Astronomical Society elected her and Caroline Herschel honorary members, the first women it admitted. In 1835 the government of Robert Peel granted her a civil list pension of £200 a year, raised to £300 in 1837 under Lord Melbourne. Francis Chantrey sculpted a marble bust of her for the Royal Society. Women could not become Fellows of the Royal Society.

In these years she was a friend and mentor to Ada, daughter of Lord Byron, later Ada Lovelace. She helped Ada with mathematics, and the two shared an interest in Charles Babbage’s engines.

Italy, last books, legacy, 1838–1872

In 1838, for the sake of William’s health and their finances, the Somervilles went to Italy for the winter. They never settled in Britain again, and Mary returned only for two brief visits. They lived in Rome, Florence, Turin, La Spezia and Naples. She went on working. The sixth edition of the Connexion, in 1842, contained her remark on the irregularities of Uranus (Lesson 08); after Neptune’s discovery, a later edition recorded that the prediction had been fulfilled. MacTutor records that her passage prompted John Couch Adams’s investigation, though Adams had already noted the problem in a memorandum of 1841.

Physical Geography appeared in 1848 and became her most successful book, used as a textbook for half a century. William Somerville died in Florence in 1860, and her son Woronzow Greig in 1865. In 1869, at the age of 88, she published On Molecular and Microscopic Science, a two-volume survey of chemistry, light and the microscopic world. The same year the Royal Geographical Society awarded her its Patron’s Gold Medal. She supported the education of women and signed John Stuart Mill’s petition to Parliament for women’s suffrage, at the head of the list.

She died in her sleep in Naples on 29 November 1872, aged 91, and was buried in the English cemetery there. Her daughter Martha edited her Personal Recollections, published in 1873.

Legacy

• Books: The Mechanism of the Heavens (1831), which brought Laplace’s celestial mechanics into English; On the Connexion of the Physical Sciences (1834, ten editions); Physical Geography (1848); On Molecular and Microscopic Science (1869).
• Firsts: with Caroline Herschel, the first women elected to the Royal Astronomical Society (1835); the occasion of the first recorded use of the word “scientist”.
• Names: Somerville Hall, founded in Oxford in 1879, later Somerville College; the lunar crater Somerville; her portrait on the Royal Bank of Scotland £10 note issued in 2017.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Mary Fairfax Somerville, MacTutor History of Mathematics.
2. Somerville, Mary, Oxford Dictionary of National Biography article reproduced by MacTutor.
3. Mary Somerville, Wikipedia (with extensive references).
4. M. Somerville, Personal Recollections, from Early Life to Old Age, of Mary Somerville, ed. M. Somerville, John Murray, 1873.
5. K. Neeley, Mary Somerville: Science, Illumination, and the Female Mind, Cambridge University Press, 2001.
6. E. C. Patterson, Mary Somerville and the Cultivation of Science, 1815–1840, Martinus Nijhoff, 1983.

Physics

1. M. Somerville, Mechanism of the Heavens, John Murray, 1831.
2. M. Somerville, On the Connexion of the Physical Sciences, John Murray, 1834; 6th ed. 1842.
3. M. Somerville, On the magnetizing power of the more refrangible solar rays, Royal Society, 1826.
4. M. Somerville, Physical Geography, John Murray, 1848; On Molecular and Microscopic Science, John Murray, 1869.
5. W. Whewell, review of On the Connexion of the Physical Sciences, Quarterly Review 51 (1834).
6. C. D. Murray and S. F. Dermott, Solar System Dynamics, Cambridge University Press, 1999 (disturbing function, resonances, the great inequality).
7. W. M. Kaula, An Introduction to Planetary Physics, Wiley, 1968 (figure of the Earth, Clairaut’s theorem).
8. D. E. Cartwright, Tides: A Scientific History, Cambridge University Press, 1999.