The Crystal Ball of James MacCullagh

Dublin University Mathematical Society

7:30pm Thursday 25th March 2010

Maxwell Lecture Theatre

Nigel Buttimore

Trinity College Dublin

James MacCullagh was born in the townland of Landahussy, Upper Badoney, County Tyrone in 1809. He entered Trinity College Dublin aged 15 years, became a Fellow in 1832, Professor of Mathematics in 1834 and later Professor of Natural and Experimental Philosophy from 1843 to 1847.

In 1838 James MacCullagh was awarded the Royal Irish Academy's Cunningham Medal for his work 'On the laws of crystalline reflexion'. In 1842, a year prior to his becoming a Fellow of the Royal Society, London, he was awarded their Copley Medal for his work 'On surfaces of the second order'.

James MacCullagh played a key role in building up the Academy's collection of Irish antiquities, now housed in the National Museum of Ireland. Although not a wealthy man, he purchased the early 12th century Cross of Cong, using what was at that time, his life savings.

He died on 24 October 1847 and is buried at St Patrick's Church, Upper Badoney, County Tyrone, where a plaque was unveiled on 15 May 2009 to honor the bicentenary of the birth of James MacCullagh.

In the section on Oscillations, Waves, and Hilbert Space of the first Chapter on Classical Mechanics of his book Mathematical Foundations of Quantum Mechanics, George Mackey considers waves in an infinite elastic medium and how they relate to the electromagnetic theory of light

the work of Young and Fresnel between 1801 and 1827 suggested that light consisted of transverse waves only
… … …
the space through which light travels is most unlike an elastic solid in which longitudinal waves must always be produced when transverse waves cross the boundary from one medium to another
… … …
No experimental evidence for longitudinal light waves could be found
… … …
From a mathematical point of view James MacCullagh showed in his Crystalline Reflection and Refraction paper of 1839 that a potential energy of the form

½  ∂yj εijk  akℓ   εℓmn yn 

with Einstein summation over repeated indices in the set {1, 2, 3}, the yj  referring to the displacement at each location xi  and time t, together with a kinetic energy

½  ∂yi  ∂yi

for an elastic solid would behave in all respects as light waves were observed to do. MacCullagh's potential differs from that of an elastic solid in utilising, instead of the components of the symmetric part, the components of the antisymmetric part of the displacement tensor


See also the paper by George Green in the Transactions of the Cambridge Philosophical Society of 1838, The Maxwellians by Bruce J. Hunt (2005) and Scientific realism: how science tracks truth by Stathis Psillos (1999).

The variational principle of Hamilton indicates that an integral over a space-time region of the Lagrangian provides the dynamical equations for light. A variation of the kinetic energy, for example, gives an integral over

δ ( ½  ∂y ∂yj )  =  ( y) δ ( y
=  ( y) ∂( δ y

and integration by parts here transfers the derivative from the second term to the first term yielding a space-time integral of

δ ( ½  ∂y ∂yj )  =  0  −  ( t2y) δ y

upon neglecting a surface term involving zero variations on the boundary. In a similar way, variation of the potential energy term leads to a space-time integral over

δ (½ ∂yj εijk akℓ εℓmn y) =  −  ( εijk akℓ  εℓmn y) δ y

and variation of the Lagrangian, the difference between kinetic and potential energies, namely a space-time integral of

½  δ ( ∂yj ∂yj   −  ∂yj εijk akℓ  εℓmn y)

enabled MacCullagh to derive in 1839 the following dynamical equations for light, essentially wave equations with the diagonal elements of the rotationally diagonalised matrix formed from the constant coefficients aij, namely aii with index i not summed, representing the squares of the velocity of the waves along the three mutually orthogonal directions of the optic axes

t2yj   =  εijk akℓ  εℓmn  ∂y.

In 1878, G. F. Fitzgerald noticed that the expression for the energy discovered by James MacCullagh, namely,

½ ( ∂yj ∂yj  +  ∂yj εijk akℓ  εℓmn y)

may be put in a form familiar from Maxwell's electrodynamics of 1865 when the time derivative of the displacement yi is identified with the magnetic field Hi and when the curl of the displacement yi involving antisymmetric differences of the spatial derivatives of yi is identified with the electric displacement Di where Di = μ aij Ej is related to the electric field Ej via dielectric constants which may have different values along orthogonal optic axes in the case of anisotropic materials. Multiplying MacCullagh's energy expression by the magnetic permeability constant μ, where the magnetic induction Bi obeys Bi = μ Hi, provides the energy function

½ ( BHi  +  EDi ) .

The analytical form of the Lagrangian found by MacCullagh, and thus the form of the energy, is therefore very modern. In a quantised form, it appears in the electrodynamical part of the Lagrangian of the Standard Model of fundamental interactions

(1/4) Fab Fba    

At a Joint event organised by HMI, RIA, TCD School of Mathematics, TCD School of Physics, National Museum of Ireland and Glenelly Historical Society, County Tyrone, "Geometry & Physics: James MacCullagh (1809-1847), Life & Achievements" Olivier Darrigol (Paris/Berkeley) gave an address, part of which is provided here.

An Optical Route to Maxwell's Equations?

Basic ideas: transverse waves, correlation between polarization and velocity of propagation.

Assumption: The propagation velocity (v) of transverse plane waves only depends on the direction of vibration.

Given an orientation of the plane wave, the two possible directions of the vibration are those for which the velocity is a maximum or a minimum.

Hence, the vector n normal to the wave planes and of length 1/v must belong to a fourth-degree, double-sheeted index surface.

MacCullagh's First Memoir (1830)

MacCullagh gave a geometrical explanation of Fresnel's wave surface by introducing the notion of reciprocal surfaces. He derived from the ellipsoidal index surface, its reciprocal, the ellipsoidal wave surface of Fresnel.

When the elliptic section in Fresnel's prescription became a circle, Hamilton showed in 1832 that there were an infinite number of refracted rays forming a cone of light. Conical Refraction was verified by Humphrey Lloyd and was demonstrated by James Lunney at the joint meeting. It was easy to analyse in terms of MacCullagh's geometrical approach.

Fresnel determined the relative amplitude of incident and reflected waves in the isotropic case through the boundary conditions: equality of the elastic constants on both sides (different densities), equality of the parallel components of the vibration, equality of the energy fluxes. These formulae are empirically valid if the vibration is perpendicular to the plane of polarization.

MacCullagh's Boundary Conditions (1835-1837)

In the isotropic case, he retrieved Fresnel's formulas under the assumptions: equalities of the densities on both sides (different elasticities), continuity of the vibration, and equality of the energy fluxes. The vibrations must then be in the plane of polarization.

In 1837 MacCullagh worked out the consequences of these conditions in the anisotropic case and found laws agreeing with Brewster's and Seebeck's experiments. Polar-plane theorem: "beautiful" (FitzGerald), "remarkably elegant" (Poincaré). This is the first published, complete theory of refraction for anisotropic media. Neumann obtained similar results earlier but published them later. For this work, MacCullagh received a medal of the Royal Irish Academy and Hamilton's praise:

It may well be judged a matter of congratulation when minds endowed with talents so high as those which Mr. MacCullagh possesses are willing to apply them to the preparatory but important task of discovering, from the phenomena themselves, the mathematical laws which connect and represent those phenomena, and are in a manner intermediate between facts and principles, between appearances and causes.

Of his own view of the work on anisotropic refraction MacCullagh wrote (page 129 of Collected Works)

If we are asked what reasons can be assigned for the hypotheses on which the preceding theory is founded, we are far from being able to give a satisfactory answer. We are obliged to confess that, with the exception of the law of vis viva, the hypotheses are nothing more than fortunate conjectures. These conjectures are very probably right, since they have led to elegant laws which are fully borne out by experiments; but this is all that we can assert respecting them.

We cannot attempt to deduce them from first principles; because, in the theory of light, such principles are still to be sought for. It is certain, indeed, that light is produced by undulations, propagated, with transversal vibrations, through a highly elastic ether; but the constitution of this ether, and the laws of its connexion (if it has any connexion) with the particles of bodies, are utterly unknown.

The peculiar mechanism of light is a secret that we have not yet been able to penetrate …. In short, the whole amount of our knowledge, with regard to the propagation of light, is confined to the laws of phenomena: scarcely any approach has been made to a mechanical theory of those laws ….

But perhaps something might be done by pursuing a contrary course; by taking those laws for granted, and endeavouring to proceed upwards from them to higher principles.

The Feynman Lectures II-1-9: the correct equations for the behavior of light in crystals were worked out by McCullough in 1843. …  If people had been more open-minded, they might have believed in the right equations for the behavior of light a lot earlier than they did.

Whittaker 1910: He succeeded in placing his own theory [of the refraction of light by crystals] on a sound dynamical basis; thereby effecting that reconciliation of the theories of light and dynamics which had been the dream of every physicist since Descartes.

Roger Penrose (The Road to Reality, Jonathan Cape, 2004, page 441)

“A profound shift in Newtonian foundations had already begun in the 19th century, before the revolutions of relativity and quantum theory in the 20th. The first hint that such a change might be needed came from the wonderful experimental findings of Michael Faraday in about 1833, and from the pictures of reality that he found himself needing in order to accommodate these.

Basically, the fundamental change was to consider that the ‘Newtonian particles’ and the ‘forces’ that act between them are not the only inhabitants of our universe. Instead, the idea of a ‘field’ with a disembodied existence of its own was now having to be taken seriously.

It was the great Scottish physicist James Clerk Maxwell who, in 1864, formulated the equations that this ‘disembodied field’ must satisfy, and he showed that these fields can carry energy from one place to another. These equations unified the behaviour of electric fields, magnetic fields, and even light, and they are now known simply as Maxwell's equations, the first of the relativistic field equations.

From the vantage point of the 20th century, when profound advances in mathematical technique have been made (and here I refer particularly to the calculus on manifolds  …”), Maxwell's equations seem to have a compelling naturalness and simplicity that almost make us wonder how the electric/magnetic fields could ever have been considered to obey any other laws.

But such a perspective on things ignores the fact that it was the Maxwell equations themselves that led to a very great many of these mathematical developments. It was the form of these equations that led Lorentz, Poincaré, and Einstein to the spacetime transformations of special relativity which, in turn, led to Minkowski's conception of spacetime.

In the spacetime framework, these equations found a form that developed naturally into Cartan's theory of differential forms; and the charge and magnetic flux conservation laws of Maxwell's theory led to the body of integral expressions that are now encapsulated so beautifully by that marvellous formula referred to as the fundamental theorem of exterior calculus. ”

Perhaps, in seeming to attribute all these advances to the influence of Maxwell's equations, I have taken a somewhat too extreme position with these comments. Indeed, while Maxwell's equations undoubtedly had a key significance in this regard, many of the precursors of these equations, such as those of Laplace, d'Alembert, Gauss, Green, Ostrogradski, Coulomb, Ampère, and others have also had important influences.

Yet it was still the need to understand electric and magnetic fields that largely supplied the driving force behind these developments—these, and the gravitational field also.

Roger Penrose (The Road to Reality, Jonathan Cape, 2004, page 491) offers a dissident view:

“However, I must confess my unease with this as a fundamental approach. I have difficulties in formulating my unease, but it has something to do with the generality of the Lagrangian approach, so that little guidance may be provided towards finding the correct theories.

Also the choice of Lagrangian is often not unique, and sometimes rather contrived—even to the extent of undisguised complication. There tends to be a remoteness from actual ‘hands‑on’ understanding, particularly in the case of Lagrangians for fields. … 

Lagrangians for fields are undoubtedly useful as mathematical devices, and they enable us to write down large numbers of suggestions for physical theories. But I remain uneasy about relying upon them too strongly in our searches for improved fundamental physical theories.”

Given a charge density ρ(t, xi) and 3‑current density Ji(t, xi) at time t and location xi in a space with three spatial dimensions where i ∈ {1, 2, 3}, the Maxwell equations of electrodynamics, in Heaviside-Lorentz units, are

c εijk Bk   −  ∂Ei  =  Ji ,      ∇Ei   =  ρ ,
c εijk Ek   +  ∂Bi  =  0 ,      ∇Bi   =  0 ,

i here referring to ∂/∂x and ∂t referring to ∂/∂t, c being the limiting velocity of electrodynamics and εijk the totally antisymmetric Levi-Civita symbol. In Minkowski space-time, the Maxwell equations take the form

c gμρ μ Fρσ  =  Jσ ,
εμνρσ μ Fρσ  =  0σ ,

from the first equation of which follows conservation of charge σ Jσ = 0 , and from the second equation, the Bianchi identity

μ Fρσ  +  ∂ρ Fσμ  +  ∂σ Fμρ  =  0 

When asked about the impact of the French Revolution of 1789 Premier Zhou Enlai of China (1949-1976) said “It is too early to tell.” One consequence though, the death of Ampère's father in Lyon, probably resulted in Ampère moving to Paris in 1805 to develop the science of electrodynamics upon hearing of H. C. Ørsted’s discovery in 1820 that magnetism and electricity were related.

Is it too soon to say what the impact of MacCullagh's work may be? He clearly was a man ahead of his time and perhaps it is too soon. Time will tell.

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