John Herschel

How to use this course

John Herschel was the most admired British man of science of the mid-nineteenth century. He is remembered by astronomers for his catalogues of nebulae and double stars, by historians of photography for the fixer and the words “negative” and “positive”, and by philosophers for an influential account of scientific method. Mathematicians know him for the notation \(f^{-1}\) and for a theorem on finite differences that still carries his name.

This course follows the man and the work together. Part II takes six of his scientific contributions and treats each as a short technical lesson: the algebra of functions, the operator calculus of differences, the theory of lens aberrations, the computation of binary-star orbits, the derivation of the normal law of errors, and the photochemistry behind early photography.

Each technical lesson opens with an italic paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions.

Timeline

Slough, 1792–1809

John Frederick William Herschel was born at Slough, near Windsor, on 7 March 1792. His father, William Herschel, was a German-born musician turned astronomer who had discovered Uranus in 1781 and was then building the largest telescopes in the world. William was 55 when his son was born. His mother, Mary Pitt, was a wealthy widow. John was their only child.

The household at Observatory House was a working observatory. His aunt Caroline Herschel, herself a discoverer of comets and her brother’s indispensable assistant, helped to raise him and gave him early lessons in experiment. The boy grew up among mirrors, eyepieces and the record books of the great sweeps of the sky in which William and Caroline catalogued some 2,500 nebulae and star clusters.

His schooling was uneven. After a local school at Hitcham he went to Eton at about eight, where he was bullied, and his mother withdrew him within months. He was then taught at home and at Clewer, including mathematics from a private tutor named Rogers. By the time he went to Cambridge he read Continental mathematics with ease.

Cambridge and the Analytical Society, 1809–1816

Herschel entered St John’s College, Cambridge, in 1809. English mathematics was then isolated. Cambridge taught the calculus in Newton’s fluxional dot notation, and few students read the analysis of Euler, Lagrange and Laplace. With Charles Babbage and George Peacock he founded the Analytical Society in 1812 to change this. Babbage joked that they stood for the pure “D-ism” of Leibniz’s notation against the “dot-age” of the university.

The Society published a volume of Memoirs in 1813, most of it written by Herschel and Babbage. In 1816 the three friends issued an English translation of Sylvestre François Lacroix’s elementary treatise on the differential and integral calculus. In 1820 they followed it with two volumes of worked examples; Herschel wrote the part on finite differences. These books, and Peacock’s later reforms of the Tripos examination, brought the Leibnizian calculus into the English curriculum within a generation.

In January 1813 Herschel graduated as Senior Wrangler, with Peacock second, and took the first Smith’s Prize. He became a fellow of St John’s, and the same year he was elected a Fellow of the Royal Society on the strength of a paper applying Cotes’s theorem. That paper introduced the inverse-function notation studied in Lesson 04.

From law to the telescope, 1814–1833

His father hoped he would enter the Church. Herschel chose the law instead and moved to London in February 1814. The experiment lasted about eighteen months. He returned to Cambridge as a tutor and examiner, and after a summer with his ageing father in 1816 he decided to devote himself to astronomy. He would carry on William’s survey of the sky, re-examining the double stars and nebulae with telescopes he ground and polished himself.

The 1820s made him the leading figure of British science. He helped found the Astronomical Society of London in 1820 (the Royal Astronomical Society from 1831) and later served three terms as its president. He received the Copley Medal in 1821. With James South he measured hundreds of double stars; their catalogue appeared in the Philosophical Transactions for 1824, and the Paris Academy awarded the Lalande Prize in 1825. He was Secretary of the Royal Society from 1824 to 1827. Travels with Babbage took him to Paris, where they met Laplace, François Arago and Jean-Baptiste Biot, and to Munich, where he visited Joseph von Fraunhofer.

On 3 March 1829 he married Margaret Brodie Stewart at St Marylebone Church in London. Their large family included the astronomer Alexander Stewart Herschel, the military surveyor John Herschel the younger, and Sir William James Herschel, a pioneer of fingerprint identification.

In the winter of 1830–31 his Preliminary Discourse on the Study of Natural Philosophy appeared in Dionysius Lardner’s Cabinet Cyclopaedia. It set out a method of science built on observation, induction from carefully gathered instances, bold hypothesis and severe testing. Michael Faraday and the young Charles Darwin both read it closely. In November 1830 reformers in the Royal Society put him forward for the presidency, and he narrowly lost to the Duke of Sussex, a setback that fed the movement to professionalize British science. He was knighted in 1831.

Functions, inverses and iterates

For the general reader. Every student who presses the \(\sin^{-1}\) key on a calculator uses Herschel’s notation. In 1813 he proposed treating a function as an object that can be composed with itself and undone, so that \(f^2\) means “apply \(f\) twice” and \(f^{-1}\) means “undo \(f\)”. The notation stuck, along with the confusion it causes.

The notation

In the Philosophical Transactions for 1813 Herschel wrote \(\cos^{-1}e\) for the arc whose cosine is \(e\), and warned that the symbol must not be read as \(1/\cos e\). His idea was systematic. For a function \(f\) write \(f^2(x)=f(f(x))\), \(f^n=f\circ f^{n-1}\), and let \(f^{-1}\) be the function with \(f(f^{-1}(x))=x\). Then the index laws hold:

In 1820 he acknowledged that Heinrich Bürmann had used a similar idea earlier. Trigonometry kept the older habit \(\sin^2x=(\sin x)^2\), so the modern reader lives with both conventions. Herschel’s choice is the right one for the theory of iteration and dynamical systems, where \(f^n\) always means the \(n\)th iterate.

The second statement, conjugation, was the central tool of the functional-equation papers that Babbage wrote under the Analytical Society’s influence (1815–16). To solve \(f^n=\mathrm{id}\), find one simple solution \(g\) and conjugate it by an arbitrary invertible \(\varphi\).

Cotes’s theorem

Herschel’s 1813 paper took its occasion from Roger Cotes’s factorization, which in modern form reads

It follows from the roots \(x=e^{\pm i(\theta+2\pi k/n)}\) of \(x^n=e^{\pm in\theta}\), and expresses the polynomial in terms of the cosine of a multiple angle, a natural setting for inverse trigonometric functions.

Finite differences and Herschel’s theorem

For the general reader. The calculus of finite differences studies how sequences change from one term to the next, the discrete cousin of differentiation. Herschel and his friends treated the operations “shift” and “difference” as algebraic symbols that could be added, multiplied and expanded in series. In 1816 he used this symbolic method to expand any function of \(e^t\) in powers of \(t\).

Operators

For a function \(u\) on the integers, define \(Eu(x)=u(x+1)\) and \(\Delta u(x)=u(x+1)-u(x)\), so \(\Delta=E-1\). Powers expand by the binomial theorem because \(E\) and \(1\) commute:

Write \(\Delta^k0^n\) for \(\Delta^k\) applied to \(u(x)=x^n\) and evaluated at \(x=0\). These are the “differences of nothing” that fill the tables of nineteenth-century textbooks:

The theorem

Proof. Expand \((e^t-1)^k\) by the binomial theorem and then each exponential:

Since \((e^t-1)^k=O(t^k)\), we have \(\Delta^k0^n=0\) for \(n<k\). Now substitute \(u=e^t-1\) into \(f(1+u)=\sum c_ku^k\) and collect the coefficient of \(t^n\); only \(k\le n\) contribute. \(\square\)

Herschel wrote the result compactly as \(f(e^t)=f(1+\Delta)\,e^{0\cdot t}\): expand \(e^{xt}=\sum x^nt^n/n!\), apply \(f(E)\) to each \(x^n\), and set \(x=0\). For \(f(z)=z^m\) this gives \(E^m0^n=m^n\), and the formula reduces to \(e^{mt}=\sum m^nt^n/n!\).

The combinatorial reading: \(\Delta^k0^n\) counts surjections from an \(n\)-set onto a \(k\)-set, by inclusion–exclusion over the \(k-i\) values missed. Dividing by \(k!\) forgets the labels of the blocks.

Lens aberrations and Herschel’s condition

For the general reader. A perfect lens would send every ray from one point of an object to one point of the image. Real lenses fail, and the failures are called aberrations. In 1821 Herschel analysed the aberrations of compound lenses and found a condition that an optical system must meet to image a short stretch of its axis sharply. It is now called Herschel’s condition, and it turns out to conflict with the better-known sine condition of Ernst Abbe.

Setting

A rotationally symmetric system images an axial point \(O\) in a medium of index \(n\) to an axial point \(O'\) in index \(n'\). A ray leaves \(O\) at angle \(\theta\) to the axis and arrives at \(O'\) at angle \(\theta'\). Perfect imaging of \(O\) means all rays take the same optical path length from \(O\) to \(O'\) (Fermat’s principle). We ask what else is needed for nearby points to be imaged perfectly too.

Derivation

Move the object point by a small vector \(\delta\mathbf r\) and suppose its image moves by \(\delta\mathbf r'\). To first order, the optical path along a ray with unit directions \(\mathbf s\) (object side) and \(\mathbf s'\) (image side) changes by \(-n\,\mathbf s\cdot\delta\mathbf r+n'\,\mathbf s'\cdot\delta\mathbf r'\). Perfect imaging of the displaced point requires this change to be the same for every ray.

Transverse displacement. Take \(\delta\mathbf r=\delta y\,\hat{\mathbf y}\), \(\delta\mathbf r'=\delta y'\,\hat{\mathbf y}\), and rays in the meridional plane. The change is \(-n\,\delta y\sin\theta+n'\,\delta y'\sin\theta'\), which vanishes for the axial ray, so it must vanish for all:

Axial displacement. Take \(\delta\mathbf r=\delta z\,\hat{\mathbf z}\), \(\delta\mathbf r'=\delta z'\,\hat{\mathbf z}\). The change is \(-n\,\delta z\cos\theta+n'\,\delta z'\cos\theta'\), and it must equal its value \(-n\,\delta z+n'\,\delta z'\) on the axis. Hence \(n\,\delta z(1-\cos\theta)=n'\,\delta z'(1-\cos\theta')\), that is,

In the paraxial limit this reads \(n\,\delta z\,\theta^2=n'\,\delta z'\,\theta'^2\). The Lagrange invariant \(ny\theta=n'y'\theta'\) gives \(\theta'/\theta=n/(n'M)\), so the longitudinal magnification is \(\delta z'/\delta z=(n'/n)M^2\). Substituting back:

Why they conflict. Dividing the two conditions gives \(\sin\theta/\sin\frac\theta2=\sin\theta'/\sin\frac{\theta'}2\), i.e. \(\cos\frac\theta2=\cos\frac{\theta'}2\). So a system with \(M\ne n/n'\) cannot image a small volume around \(O\) perfectly. The microscope designer must choose; Abbe’s choice, the sine condition, governs objectives because their job is to image a plane.

Double stars and binary orbits

For the general reader. William Herschel discovered that some close pairs of stars orbit each other, proving that Newton’s gravity works beyond the solar system. His son measured thousands of such pairs and, in 1833, gave astronomers a practical way to turn rough measurements into orbits. From an orbit and a distance one obtains the masses of the stars, still the most direct way to weigh a star.

The apparent orbit

Observers measure the separation \(\rho\) (in arcseconds) and position angle \(\theta\) of the fainter star relative to the brighter. The relative orbit is a Kepler ellipse in a plane tilted to the sky, and its projection onto the sky is again an ellipse, the apparent orbit. The primary star lies at a focus of the true orbit but generally not at a focus of the apparent one.

Herschel’s method (1833)

Separations measured with a filar micrometer were unreliable for close pairs, while position angles were good. Herschel therefore plotted the position angles against time and drew a smooth curve by hand through the points, smoothing out observational error by eye. The slope of that curve gives \(d\theta/dt\), and the law of areas then gives the distance:

The constant \(C\) is fixed by scaling to the more reliable wide separations. Plotting \((\rho,\theta)\) yields the apparent ellipse, from which the true elements follow by geometry. Félix Savary had computed the first binary orbit, for ξ Ursae Majoris, in 1827 by an analytical method; Herschel’s graphical approach was more forgiving of poor data and was widely used.

Weighing the stars

Errors, isotropy and the normal law

For the general reader. Why do errors of measurement follow the bell-shaped curve? In 1850, reviewing Adolphe Quetelet’s work on probability, Herschel gave a short argument. Imagine throwing darts at a target. If the horizontal and vertical errors are independent, and the chance of a hit depends only on its distance from the centre, then the bell curve is the only possibility. Ten years later James Clerk Maxwell used the same argument for the velocities of gas molecules.

Proof

Write \(\phi(x)\phi(y)=g(x^2+y^2)\). Setting \(y=0\) gives \(g(x^2)=\phi(x)\phi(0)\), so \(\phi\) is even and \(\phi(0)>0\) (otherwise \(\phi\equiv0\)). Put \(h(s)=\log\big(\phi(\sqrt s)/\phi(0)\big)\) for \(s\ge0\), where \(\phi>0\). Dividing the identity by \(\phi(0)^2\) and taking logarithms,

This is Cauchy’s functional equation, and its continuous solutions are linear: \(h(s)=-\lambda s\). (One checks that \(\phi\) has no zeros, since a zero at \(x_0\) would force \(\phi(x)\phi(y)=0\) on the whole circle of radius \(\lvert x_0\rvert\).) Hence \(\phi(x)=\phi(0)e^{-\lambda x^2}\), and integrability forces \(\lambda>0\). Normalizing gives the Gaussian with \(\lambda=1/2\sigma^2\). \(\square\)

Maxwell’s use

In his 1860 paper on the kinetic theory of gases, Maxwell assumed that the three velocity components of a molecule are independent and that their joint distribution depends only on speed. The same argument in three dimensions gives the velocity distribution, and passing to the speed \(v=\lvert\mathbf v\rvert\) gives

with \(\sigma^2=kT/m\) identified later from thermodynamics. Maxwell knew Herschel’s review, which had been reprinted in Herschel’s collected essays in 1857, and historians regard it as his likely source.

Light, chemistry and photography

For the general reader. An early photograph faded as soon as it saw daylight, because the unexposed silver salts kept darkening. Herschel had found in 1819 that a salt then called hyposulphite of soda dissolves silver chloride. In 1839 he told William Henry Fox Talbot, and through others Louis Daguerre, that it would fix their images. “Hypo” is still the photographer’s word for fixer. He went on to invent the blueprint and to measure the heating power of sunlight.

Fixing

Light reduces some silver halide grains to metallic silver; the rest remain sensitive. Thiosulfate (Herschel’s hyposulphite) removes them by forming soluble complexes:

The reaction does not attack metallic silver, so the image survives while the unexposed salt is washed away. In early 1839 Herschel also made one of the first photographs on glass, and in papers to the Royal Society in 1839, 1840 and 1842 he introduced the terms “negative” and “positive” and helped popularize the word “photography”.

The cyanotype (1842)

Because the process needs only iron salts and water, it was cheap and permanent. Anna Atkins used it from 1843 for her books of botanical impressions, and engineers used it as the “blueprint” until the mid-twentieth century.

The actinometer

Herschel’s actinometer of 1825 measured the heating power of sunlight. A thermometer bulb in a vessel of dark liquid was exposed to the Sun and then shaded, alternately, for equal intervals. Comparing the warming rates in sun and shade cancels the losses to the surroundings, to first order. He used it at the Cape and elsewhere to study solar radiation and its absorption by the atmosphere.

Star gauges and the shape of the Galaxy

For the general reader. How can one map the Milky Way without knowing the distance to a single star? William Herschel’s answer was to count. If stars are spread evenly and the telescope reaches the edge of the system, then more stars in a field means the system extends further in that direction. At the Cape his son repeated these “star gauges” over the southern sky, counting stars in thousands of telescope fields, and found the Milky Way to be a flattened, irregular disc seen from inside.

Depth from counts

Suppose stars have uniform number density \(\nu\) and the telescope sees every star out to the boundary of the system, at distance \(R\) in a given direction. A field of solid angle \(\omega\) is a cone containing

So the gauge number fixes the relative depth: a field with eight times as many stars reaches twice as far.

Proof. Flux falls as \(F=L/4\pi r^2\), and \(m=-2.5\log_{10}F+\text{const}\). A star is brighter than \(m\) when \(r<r_m\) with \(r_m\propto10^{0.2m}\). The count is \(N\propto r_m^3\propto10^{0.6m}\). For a mixture of luminosities each class obeys the same law with its own constant. \(\square\)

Observed counts grow more slowly than this at faint magnitudes. Herschel and his father read this as the edge of the system. The real cause is partly interstellar absorption by dust, not understood until the 1930s, which hid the true size of the Galaxy and the Sun’s off-centre place in it.

The Cape and the Mint, 1833–1855

To complete his father’s survey he needed the southern sky. At his own expense, Herschel sailed from Portsmouth on 13 November 1833 with his wife, three children and his 20-foot reflector, and reached Cape Town on 15 January 1834. From the estate of Feldhausen at Claremont he swept the southern heavens for four years. He catalogued about 1,700 nebulae and clusters and over 2,000 double stars, counted stars in thousands of fields to map their distribution, observed Halley’s comet on its 1835 return, and watched the great outburst of η Carinae from December 1837. He also mapped the Magellanic Clouds, measured solar radiation, and botanized with his wife, who drew the Cape flora.

On 3 June 1836 the Beagle called at the Cape and Charles Darwin and Robert FitzRoy visited him. Earlier that year Herschel had written to Charles Lyell calling the replacement of extinct species by new ones the “mystery of mysteries”, a phrase Darwin later used in the opening of On the Origin of Species. Herschel never accepted natural selection as a sufficient explanation.

He returned to England in 1838 to public acclaim and was made a baronet at the coronation of Queen Victoria. The reduction of the Cape observations took nine years. Results of Astronomical Observations made at the Cape of Good Hope appeared in 1847 and won his second Copley Medal. The same year he proposed the names still used for the seven then-known moons of Saturn (Mimas, Enceladus, Tethys, Dione, Rhea, Titan and Iapetus), and in 1852 the names Ariel, Umbriel, Titania and Oberon for the moons of Uranus. His Outlines of Astronomy (1849) was the standard English text for decades and introduced Julian day numbers into general astronomical use.

In 1850 he accepted the post of Master of the Mint, the office Newton had held. The work was administrative and exhausting, and it kept him from research. His health suffered, and he resigned in 1855.

Collingwood, death and legacy, 1855–1871

Herschel spent his last sixteen years at Collingwood, his house near Hawkhurst in Kent. He completed the great catalogues. The General Catalogue of Nebulae and Clusters of Stars, published in the Philosophical Transactions for 1864, brought together his father’s discoveries, his own, and those of others in about 5,000 entries. John Louis Emil Dreyer’s New General Catalogue of 1888 grew directly out of it, which is why objects still carry NGC numbers. A companion catalogue of more than 10,000 double and multiple stars was published after his death, in 1874. He also translated Homer’s Iliad into English verse and wrote widely read essays on science for the reviews.

He died at Collingwood on 11 May 1871, aged 79. He received a national funeral and was buried in Westminster Abbey, close to the grave of Newton. The Proceedings of the Royal Society judged his loss the greatest to British science since Newton’s death.

Legacy

• Mathematics: the notation \(f^{-1}\); the symbolic calculus of operators, carried on by George Boole; Herschel’s theorem on differences of zero; and the transformation of Cambridge mathematics with Babbage and Peacock.
• Astronomy: the first systematic survey of the southern sky, the General Catalogue behind the NGC, the graphical method for binary orbits, and the names of the moons of Saturn and Uranus.
• Physics: Herschel’s condition in optics, the actinometer, and the isotropy argument for the normal law that Maxwell carried into kinetic theory.
• Photography: the thiosulfate fixer, the cyanotype, and the basic vocabulary of the art.
• Method: the Preliminary Discourse shaped the views of William Whewell, John Stuart Mill and Darwin on how science proceeds.
• Honours: two Copley Medals, Royal Medals in 1836 and 1840, two Gold Medals of the Royal Astronomical Society, the Lalande Prize, a baronetcy; lunar crater J. Herschel and Martian crater Herschel.

Connections

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

Influenced by

• Joseph-Louis Lagrange
• Pierre-Simon Laplace

Influenced

• George Boole
• George Gabriel Stokes
• James Clerk Maxwell
• Charles Darwin 1836

Colleagues and collaborators

• Augustin-Jean Fresnel
• David Brewster
• François Arago
• Friedrich Wilhelm Bessel
• George Biddell Airy
• Jean-Baptiste Biot
• Urbain Le Verrier
• George Peacock 1812-1820
• James South 1821-1824
• William Henry Fox Talbot 1824-1871

Friends

• Charles Babbage 1810-1871
• Mary Somerville
• Thomas Young

Family

• Caroline Herschel
• William Herschel
• Alexander Stewart Herschel

Also mentioned in this life

• Augustin-Louis Cauchy
• Leonhard Euler
• Michael Faraday
• Pierre de Fermat
• Joseph von Fraunhofer
• Carl Friedrich Gauss
• Edmond Halley
• Johannes Kepler
• Gottfried Wilhelm Leibniz
• August Ferdinand Möbius
• Isaac Newton
• Lord Rayleigh

Fields

• astronomy 50 lives
• mathematics 15 lives
• optics 38 lives
• philosophy of science 7 lives

On CoursesHub

• Astrophysics for Beginners
• Optics & Photonics
• Philosophy of Science

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, John Frederick William Herschel, MacTutor History of Mathematics.
2. John Herschel, Wikipedia (with extensive references).
3. G. Buttmann, The Shadow of the Telescope: A Biography of John Herschel, Scribner, 1970.
4. S. Ruskin, John Herschel’s Cape Voyage, Ashgate, 2004.
5. D. S. Evans et al. (eds.), Herschel at the Cape: Diaries and Correspondence 1834–1838, University of Texas Press, 1969.

Mathematics and science

1. J. F. W. Herschel, On a remarkable application of Cotes’s theorem, Phil. Trans. R. Soc. 103 (1813) 8–26.
2. J. F. W. Herschel, On the development of exponential functions; together with several new theorems relating to finite differences, Phil. Trans. R. Soc. 106 (1816).
3. J. F. W. Herschel, On the aberrations of compound lenses and object-glasses, Phil. Trans. R. Soc. 111 (1821). See Herschel’s condition.
4. J. F. W. Herschel, On the investigation of the orbits of revolving double stars, Mem. R. Astron. Soc. 5 (1833).
5. J. F. W. Herschel, Quetelet on probabilities, Edinburgh Review 92 (1850) 1–57. See Maxwell’s theorem.
6. J. F. W. Herschel, On the action of the rays of the solar spectrum on vegetable colours, and on some new photographic processes, Phil. Trans. R. Soc. 132 (1842).
7. J. F. W. Herschel, A Preliminary Discourse on the Study of Natural Philosophy, Longman, 1830; Outlines of Astronomy, 1849.
8. J. F. W. Herschel, A general catalogue of nebulae and clusters of stars, Phil. Trans. R. Soc. 154 (1864).
9. W. Bryc, Maxwell and the normal distribution, arXiv:1702.01411 (on the history of the isotropy argument).
10. On two applications of Herschel’s theorem, arXiv:1205.1104.
11. F. Cajori, A History of Mathematical Notations, vol. 2, Open Court, 1929 (on \(f^{-1}\)).
12. G. Boole, A Treatise on the Calculus of Finite Differences, Macmillan, 1860.