Sir William Rowan Hamilton was an Irish mathematician, Andrews professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Ireland. Hamilton’s work is highly praised; below a short overview is given of the still growing influence of his work.
Yet in stark contrast to how his work is seen is the very negative contemporary view on his private life. Although Hamilton lived a for his time and social status overall quite normal private life, the first seeds for the negative later views already emerged in his lifetime. That story is too long to fully discuss here, but I extensively discussed it in my 2017 book A Victorian marriage of which a short overview is given below, and in my 2019 presentation, given at the Maynooth Conference in the History of Mathematics, A biographer’s opinion as primary source : the strange case of Sir William Rowan Hamilton. After Hamilton’s death the gossip evolved further, and especially of Lady Hamilton’s reputation soon nothing was left. It can easily be stated that there are very few famous couples about whom the posthumous view worsened so continuously and steadily over such a long period of time.
In 2018, in a peer-reviewed article A most gossiped about genius: Sir William Rowan Hamilton, written with Steven Wepster, we showed where the gossip originated, and how it grew over the years. Using six authors spread over a period of one hundred and seven years, we showed how details were left out and new ones introduced, which in turn were treated as facts, until of the original stories hardly anything was left.
Unfortunately, it seems that almost anything can be added to the story of Hamilton’s private life as long as it fits the caricature of an unhappy and unworldly genius, and there is no reason to assume that in the future no new details will be added to the gossip. Hamilton's life must therefore be reconsidered, and told anew. This website is dedicated to that, by discussing details from Hamilton’s private life as it can be inferred from his correspondences. They show Hamilton as a man of his time, as a happily married family man, and a very hard-working mathematician.
I started this website when I was writing my essay A Victorian Marriage - Sir William Rowan Hamilton. On the first page, VictorianMarriage.html, my publications about Hamilton are listed, and every now and then I write about new details or new insights. A second page, HamiltonPhotopage.html, is filled with photographs of Hamilton’s family, friends and colleagues, to get some notion of what people looked like in those days, placing the Hamilton photographs into perspective. A third page, CatherineDisney.html, is dedicated to the very unhappy life of Hamilton’s first love Catherine Disney; I wrote my second book about her. On the one hand to show that it was not at all strange that Hamilton was periodically upset about her fate and that that had nothing to do with his own marriage, but also because her very sad story was in itself worth to be told. Forcing people into marriages they do not want to be in should not be possible anywhere at any time. A fourth page, Miscellaneous.html, is a miscellany, partly connected to Hamilton, and partly to my former physics study, which was interrupted when I stumbled on the very strange stories which were told about the Hamiltons.
This website is ongoing work. I have planned to write a third book, about Lady Hamilton’s correspondence, and searching for information about her I sometimes again find new details, stories, remarks, which I will give here when I find them. And in the meantime I work on sundry associated subjects, whenever I encounter them. Yet I hope that some day the view on the Hamiltons’ private life becomes as positive as they deserve, and this work will be finished. While the mathematicians and the physicists carry on the work following from Hamilton’s enormous legacy, of which a glimpse is given below.
This page contains anything not at home in the other pages
From nineteenth century Dublin and Quantum mechanics
to Mars and the Universe.
From subjects belonging to Hamilton’s story,
to my own interests and doings.
Hamilton’s first discovery was a new way to describe optics. In 1833 he wrote about his work: “I am still engaged [...] on a work which has already occupied me since I was about eighteen, in attempting, with the help of the differential or fluxional calculus, to remould the geometry of light by establishing one uniform method for the solution of all the problems deduced from the contemplation of one central or characteristic relation.”
On his webpage Hamilton’s Diabolical Legacy Mike Jeffrey writes, “In 1832 Hamilton used mathematics, that strange abstract language of symbols and axioms, to predict something truly absurd, and obviously physical nonsense. Hamilton predicted a singularity, a point where light's deterministic journey through a simple crystal broke down. In one stroke, the field of singular optics was born and a sensation began that would take 173 years to run its course. Despite prompt experimental confirmation of Hamilton's beautiful mathematical theory, the phenomenon was long hindered by controversy and misconception. Victorian mathematics contained only the initial sparks of the asymptotic techniques which would be needed to achieve a full understanding. Conical refraction entered amidst a climax in the contest between undulatory and corpuscular theories of light, entwined in the earliest roots of wave asymptotics and singular optics. The modern theory of light has its origins in Christian Huygens’ 1677 wave theory. This theory did not explain diffraction and did not account for polarisation, which favoured the corpuscular theories backed by the intellectual might of Pierre-Simon Laplace and Isaac Newton. Augustin Fresnel reversed this dominance in 1816 when he discovered the wave theories of refraction and diffraction. Describing light rays as the normals to level surfaces of some characteristic function, Hamilton’s formulation of geometrical optics married the wave theory of Fresnel with the ray method of Newton. Hamilton’s prediction continues to stretch the capability of lasers and computers of the modern age, and embodies all of the complexities of multiple-scales and singularities that plague modern science.”(3) That “even this esoteric piece of Hamilton’s work is continuing to be made use of” is shown for instance in articles about Conical Refraction in Biaxial Crystals and Conical Refraction Optical Tweezers.
Hamiltonian mechanics and the Hamiltonian
In Spring 1833 Hamilton “extended from Optics to Dynamics his algebraic method of a characteristic function.” Instead of describing mechanics by the working of forces, as Newton did, Hamilton used the energies of the systems under scrutiny. This mathematically elegant reformulation of classical mechanics is now called Hamiltonian mechanics and, roughly speaking, the energy of a system is called the Hamiltonian.
In his short article called Hamiltonians Terence Tao writes, “At first glance, the many theories and equations of modern physics exhibit a bewildering diversity: compare for instance classical mechanics to quantum mechanics, non-relativistic physics to relativistic physics, or particle physics to statistical mechanics. However, there are strong unifying themes connecting all of these theories. One of these is the remarkable fact that in all of these theories, the evolution of a physical system over time (as well as the steady states of that system) is largely controlled by a single object, the Hamiltonian of that system, which can often be interpreted as describing the total energy of any given state in that system. Roughly speaking, each physical phenomenon (e.g. electromagnetism, atomic bonding, particles in a potential well, etc.) may correspond to a single Hamiltonian H, while each type of mechanics (classical, quantum, statistical, etc.) corresponds to a different way of using that Hamiltonian to describe a physical system. [... In mathematics,] the Hamiltonians play a major role in dynamical systems, differential equations, Lie group theory, and symplectic geometry. [...] Because of their ubiquitious presence in many areas of physics and mathematics, Hamiltonians are useful for building bridges between seemingly unrelated fields, for instance in connecting classical mechanics to quantum mechanics, or between symplectic mechanics and operator algebras.”(1)
Hamiltonian mechanics in mathematics
One of the examples of the use of Hamiltonian mechanics in mathematics is symplectic geometry. This is the theory of symplectic structures on manifolds, spaces that locally look like the space we live in but may have any dimension. Describing the state of a classical mechanical system by its generalized position q and generalized momentum p, phase space consists of all such pairs (q,p). In Hamilton’s formalism the motion of a classical mechanical system is governed by Hamilton’s equation, a first order ordinary differential equation for a point in phase space as a function of time. Symplectic geometry has its roots in this reformulation of classical mechanics; in Hamilton’s equation of motion, which is equivalent to Newton’s equations of motion, the standard symplectic structure occurs. Informally, a symplectic structure on a smooth manifold is an assignment that yields a real number for every 2-dimensional subset that is contained in the manifold. Symplectic geometry studies local and global properties of symplectic structures and Hamiltonian systems. A famous conjecture by Vladimir Arnold states that under certain conditions periodic solutions of Hamilton’s equation exist. Physically, such a solution corresponds to a mechanical system that returns to its initial point in phase space after a fixed time.(4)
Hamilton’s second discovery was the extension of the imaginary numbers in such a way that rotations in three dimensions could be described. Because they appeared to have four components Hamilton called them quaternions. He developed them as pure mathematics, but at the same time he aimed, and succeeded, to use them to solve physics, engineering and astronomical problems.*
The quaternions had a difficult start, because they ended commutativity in algebra which was not easily accepted. Fiacre Ó Cairbre writes, “The mathematical world was shocked at his audacity in creating a system of “numbers” that did not satisfy the usual commutative rule for multiplication. Hamilton has been called the “Liberator of Algebra” because his quaternions smashed the previously accepted convention that a useful algebraic number system should satisfy the rules of ordinary numbers in arithmetic. His quaternions opened up a whole new mathematical landscape in which mathematicians were now free to conceive new algebraic number systems that were not shackled by the rules of ordinary numbers in arithmetic. Hence, we may say that Modern Algebra was born on October 16, 1843 on the banks of the Royal Canal in Dublin.”(5) But Hamilton’s work was also regularly regarded with disbelief; in 1853 he wrote to a friend, “You will I hope bear with me if I say, that it required a certain capital of scientific reputation, amassed in former years, to make it other than dangerously imprudent to hazard the publication of [the Lectures in Quaternions] which has, although at bottom quite conservative, a highly revolutionary air. It was a part of the ordeal through which I had to pass, an episode in the battle of life, to know that even candid and friendly people secretly, or, as it might happen, openly, censured or ridiculed me, for what appeared to them my monstrous innovations.” Another problem was that quaternions live in four dimensions and therefore are hardly visualizable. Hamilton died believing that the quaternions would be at the heart of physics, but about twenty years after his death vector analysis was directly derived from the quaternions,(6) and because contrary to quaternions vector analysis was very visual, intuitive and easy to work with, in physics the quaternions themselves disappeared for a long time. Although it can easily be claimed that, in the form of vector analysis, they are at the heart of physics, just as Hamilton had expected.
Quaternions in space travel
For the quaternions the coming of computers changed everything. In the late 1950s the quaternions re-emerged, apparently at first in aircraft simulation, because computers appeared to work with quaternions more easily than with vector analysis, therewith using less computing power.** After being used in the Space Shuttle program, their use in space travel became very widespread. Noel Hughes writes, “Vectors and quaternions, along with algebra, trigonometry, geometry, etc. are an intertwined set of tools which can be used to solve many problems that can appear intractable. [Quaternions are used in] spacecraft guidance, navigation and control engineering. [...] Prior to the dawn of the space age and the need for attitude description that does not have singularities, ambiguities, etc., [for most people] quaternions were little more than an esoteric mathematical oddity. [Nowadays] quaternions are used extensively in the Aerospace industry, in the animated film industry and, to a lesser extent, in the medical world. [...] The fundamentals are simple, elegant and straight forward once the superfluous stuff is stripped away.”(7)
Quaternions in animation
In 1985 the apparently earliest article introducing quaternions to computer graphics was published. In the abstract of his 1985 article Ken Shoemake writes, “Solid bodies roll and tumble through space. In computer animation, so do cameras. The rotations of these objects are best described using a four coordinate system, quaternions, as is shown in this paper.”(8) Thereafter the quaternions in this field have been developed almost beyond recognition as can be seen in a 2020 article; Manos Kamarianakis and George Papagiannakis write, “Skinned model animation has become an increasingly important research area of Computer Graphics, especially due to the huge technological advancements in the field of Virtual Reality and computer games. The original animation techniques, based on matrices for translation, rotation and dilation, are still applied as the latest GPUs allow for fast parallel matrix operations. The fact that the interpolation result of two rotation matrices does not result in a rotation matrix, forced the use of quaternions as an intermediate step. The extra transmutation steps from matrix to quaternions and vice versa, adds some extra performance to the animation but yields better results, solving problems such as the gimbal lock. Nowadays, the state-of-the-art methods for skinned model animation use dual-quaternions, an algebraic extension of quaternions. [...] Dual-quaternions handle both rotations and translations, but cannot handle dilations. [...] Conformal Geometric Algebra (CGA) is an algebraic extension of dual-quaternions, where all entities such as vertices, spheres, planes as well as rotations, translations and dilation are uniformly expressed as multivectors. The usage of multivectors allows model animation without the need to constantly transmute between matrices and (dual) quaternions, enabling dilation to be properly applied with translation.”(9)
Yet for mathematics the history of quaternions is quite different; there they never disappeared from sight. In his upcoming book Quaternion algebras, John Voight writes, “Quaternion algebras have threaded mathematical history through to the present day, weaving together advances in [many branches of mathematics, and they ...] continue to arise in unexpected ways. [...] Quaternion algebras sit prominently at the intersection of many mathematical subjects. They capture essential features of noncommutative ring theory, number theory, K-theory, group theory, geometric topology, Lie theory, functions of a complex variable, spectral theory of Riemannian manifolds, arithmetic geometry, representation theory, the Langlands program – and the list goes on. Quaternion algebras are especially fruitful to study because they often reflect some of the general aspects of these subjects, while at the same time they remain amenable to concrete argumentation. Moreover, quaternions often encapsulate unique features that are absent from the general theory (even as they provide motivation for it). [...] The enduring role of quaternion algebras as a catalyst for a vast range of mathematical research promises rewards for many years to come.”(2)
Perhaps the most ubiquitous use of Hamilton’s work is made in astrophysics. Andrew Hamilton (no relation to Sir WRH) writes, “Astrophysicists, who harness mathematics and physics to study astronomical phenomena, use Hamilton's work every day all the time. Hamiltonians are at the core of the two pillars of modern physics, general relativity and quantum field theory. Hamiltonian dynamics is central for understanding the dynamics of gravitating systems such as solar systems and galaxies, or the dynamics of electrodynamic systems such as plasmas. Hamilton's quaternions are by far the most powerful and elegant way to understand and encode spatial rotations, and his biquaternions, having complex numbers as their coefficients, do the same thing for spacetime rotations, also called Lorentz transformations. Hamilton's impact on theoretical astrophysics is everywhere.”(10)
(1)Terence Tao (2008), Hamiltonians.
* Physics as we know it now did not exist yet, see below for the state of science in Hamilton’s days.
** In the Foreword to this 1958 article Alfred C. Robinson writes, “The general subject of quaternions as applied to coordinate conversions has been under investigation for approximately two years, though the bulk of the work reported here was accomplished during the last six months of 1957.” And in his abstract, “It is shown that the quaternion method is no more sensitive to multiplier errors than is the direction cosine method, and it requires nearly 30 per cent less computing equipment. [...] By every important criterion, the quaternion method is no worse than, and in most cases, better than the direction cosine method.”
The Dublin and Kingstown Railway was Ireland's first railway.
Copied from the Dublin Penny Journal of September 1834.
This engraving, of the steam locomotive Hibernia, built in 1834, shows the state of science a year after Hamilton’s marriage, two years after the publication of the Third Supplement to an Essay on the Theory of Systems of Rays which would lead to Hamiltonian mechanics,* and nine years before Hamilton found the quaternions.
Having made his discoveries in a time in which photography, telephone and radio did not exist yet, bicycles were in their infancy and people walked or travelled by horse, carriage, steam train and steamship, Hamilton’s son reported that “Sir W.” was indifferent to contemporary fame, “arising from his conviction that his belonged to a future age entirely.” Hamilton was right, yet little could he know that large parts of his work had to wait until quantum mechancis and the dawn of computers before they would start to blossom.
* For short descriptions and complete transcriptions of the original essay and the three supplements see David Wilkins, Theory of Systems of Rays.
In 2017 I published (a corrected version of) my first book, or actually a history essay because it is written as a ‘defense,’ A Victorian Marriage : Sir William Rowan Hamilton, in which I showed that the unhappy and alcoholic view on Hamilton is flawed, and how this idea emerged, unintentionally, from the enormous biography about Hamilton, written by Robert Perceval Graves and published deep within the Victorian era, vol 1 (1882), vol 2 (1885), vol 3 (1889). Reading the biography in the context of his time, a picture emerged of a man who did not just use his enormous intelligence for his work, but also for his private life. A genius in a happy marriage, with ups and downs as we all have.
The first seven chapters, Introduction, Early years, A lover, A brother, A husband, A good marriage, and Later years, are a description of Hamilton’s private life. In 1824, only nineteen years old, Hamilton fell in love with Catherine Disney. After some months of unspoken love she married someone else, and it took him six years to cope with his loss. In 1831 he fell in love with Ellen de Vere, who did not love him back, causing months of very melancholy feelings. Then, in the summer of 1832, Hamilton made a remarkable discovery; he saw that he was wasting his life on passion, and found a way to change his behaviour. His zest for life was restored, and he fell in love with Helen Bayly.
Indeed, contrarily to what has been suggested, that Hamilton always remained to love only Catherine and hence had an unhappy marriage, it is clear from a letter to a friend that he really had been in love with Ellen de Vere, writing that he had had “another affliction of the same kind and indeed of the same degree” as his love for Catherine had been. And having sent his ante-nuptial poems about Helen Bayly to Coleridge, believing that poems had to come from true feelings and highly admiring Coleridge, may serve as one of the ‘proofs’, if these are needed, that he indeed was very much in love with her. The chapters end with Hamilton’s later years and death.
Chapter 8, A lost love, is about what happened with his first and lost love Catherine Disney. It is shown that at the time of her marriage, in 1825, Hamilton had not known that Catherine had been forced to marry someone she did not love. Having been told that she was going to marry he had not seen her again; he saw her only in 1830, 1845, and in 1853 shortly before she died. Hamilton had assumed that Catherine had wanted to marry until, in 1830, he visited her, and saw that she was unhappy. About the 1845 visit nothing further is known, but in 1848 she told him in a correspondence that her marriage had been unhappy from the start. Only in 1853, some weeks before her death, she could finally tell him that she was coerced into this marriage, and that she had wanted to marry him. Each time Hamilton again learned about new details, and about how terribly unhappy Catherine was, he was understandably distressed. Which is entirely different from having loved only her his whole life.
The 9th chapter, Solemn dogged seriousness, is a description of the Hamiltons at home, as far as it could be extracted from the biography; although they had their ups and downs, as everyone has, they had a good and happy marriage. Familiar traits are discussed, and anecdotes given, as told by the eldest son, William Edwin, after his father’s death. One of these anecdotes became the direct origin of a part of the contemporary gossip; the view on Hamilton as a very disorderly man, which led to the conclusion that Lady Hamilton thus was a bad housewife, from which it was concluded that their marriage was unhappy, and that Hamilton started to drink alcohol because she did not serve dinner. Which clearly was not the case, as could already have been concluded from the anecdote itself. This chapter also contains a discussion of their illnesses, Hamilton’s gout and Lady Hamilton’s weak health. Very likely reasons are given for Lady Hamilton’s two nervous breakdowns, which directly had to do with her marriage; being a married woman in Victorian times was indeed extremely difficult. Yet both times the Hamiltons were able to solve their problems.
The 10th chapter, An occasional mastery, is about Hamilton’s alleged alcoholism. It is discussed what exactly happened, and that what Graves called “an occasional mastery”, hinting at it long before describing it while vaguely predicting doom to come, was a one-time event in 1846. It is also discussed how especially in Hamilton’s so-called ‘High Church days,’ which lasted from about 1839 until late in 1845 or very early in 1846, it was extremely unlikely that he was drinking too much. And indeed, Graves never claimed that Hamilton was an alcoholic.
In the first part of the 11th and last chapter, By no means an alcoholic, Hamilton’s use of alcohol is discussed on the basis of the DSM-5, using the information found in Graves’ biography. The conclusion was that Hamilton did not meet the DSM-5 conditions for alcoholism and that, had he lived now, he thus would not be regarded as alcoholic.
Having shown that Hamilton was not an alcoholic, it was considered how much risk there had been that he would have become one. To that aim the data from Graves' biography about when and how much alcohol Hamilton drank are used, in a worst case scenario, to fill in the AUDIT test which can be found on p. 17 of the manual published by the World Health Organization. It is designed to test what the risks are for someone, in this case Hamilton, to become an alcoholic. The overall conclusion was that, in this worst case scenario, during two periods of his life Hamilton would have had an increasing risk of becoming alcoholic, but that he never reached the higher risk levels of becoming one.
During these two periods of increasing risk Hamilton, who always remained temperate at home, sometimes drank much in public. The first period started in 1844 when after having discovered the quaternions Hamilton seemed to become overworked, and ended in 1846, with the event at the Geological Society. Thereafter Hamilton was warned by Charles Graves, brother of the biographer Graves and fellow mathematician at Trinity College Dublin, that he was ruining his Dublin reputation; Hamilton immediately changed his behaviour and became abstemious for two years. The second period started in the summer of 1848 and ended when, most likely in 1851, but in any case before the end of 1853, Hamilton was again warned by Charles Graves. Drinking wine, and sometimes much, at dinners and parties had been completely accepted until around 1840 the Temperance Movement had reached Dublin, and apparently the gossip about Hamilton’s drinking was worsening again. This second time he changed his behaviour less rigorously but lasting; although he did not abstain again, he never drank much any more.
It then is discussed briefly where Graves’ information about Hamilton’s private life came from; Graves lived in England from 1833 until 1864, thus from the time Hamilton married until one year before he died. Apart from writing letters other means of contact did not exist yet, and they did not visit each other frequently, apparently not even yearly, which means that Graves hardly saw Hamilton live his daily life. The essay is concluded by a revaluation of the two main biographies, written by Graves in the 1880s and by Thomas Hankins in 1980. It is shown that all the information these two biographies contain can be placed in a much more positive light than is done nowadays, by putting Hamilton’s life in the context of his time, and carefully noticing all the nuances Graves gave in his biography.