Great Astronomers: Hamilton
Great Astronomers in Modern English
by Sir Robert S. Ball, 1895 (paraphrased by Leslie Noelani Laurio)
To view the table of contents for the rest of this book, click here.
Sir William Rowan Hamilton, 1805-1865
Integrated Newtonian dynamics and invented a new branch of calculus related to Quaternions
William Rowan Hamilton was born right at midnight between August 3 and 4, 1805, in Dublin. His father, Archibald Hamilton, was a solicitor [a type of lawyer], and William was the fourth of his nine children. William's ancestors were mostly Irish gentlemen -- but his grandmother on his mother's side of the family was Scottish. When he was still a baby, his parents decided to let his uncle, James Hamilton, who was a clergyman in Trim, in County Meath [Ireland] educate him since he ran a school. James's sister, Sidney, lived with him, and William spent his childhood there.
In Robert Perceval Graves' book "Life of Sir William Rowan Hamilton" [three volumes on line at archive.org: 1, 2, 3], there are some letters included in which Aunt Sidney wrote to William's mother in Dublin to let her know how William was doing. They are extraordinary. When he was three years old, his aunt wrote that William was a "hopeful blade" because he was a strong little boy, which she illustrated by relating that William was able to make boys older than himself run away from him! A month later, she wrote that William was brought in to class to read the Bible out loud in front of the first graders to shame them for not reading as well as William. At three years old! His Uncle James worked hard to teach him, but his aunt said she couldn't imagine how he was able to pick up so much of what he taught because "he never stops playing and jumping about!" When he was 4 years and three months old, he had dinner with the vicar and amused everyone there by reading a book for them no matter whether the book was turned upside down, or sideways! Aunt Sidney wrote to his mother that "Willie is a most sensible little creature, but at the same time has a great deal of roguery." When he was four and five months, he went to town to visit his mother, and his mother wrote this to her sister:
'His ability to recite is astonishing. His obvious and accurate knowledge of geography is unbelievable. He can even sketch out the countries on paper with a pencil and cut them out! They're not exact, but close enough that anyone who knew the countries would recognize them. But that's nothing compared with how well he reads Latin, Greek, and Hebrew.'
Aunt Sidney wrote that as soon as Willie got back to Trim, he was eager to continue with his lessons. He wouldn't even eat his breakfast until his uncle had heard him recite his Hebrew and explained to him the importance of proper pronunciation. At age five, he went with his uncle to visit a friend and recited long passages from John Dryden. Another gentleman was there who was skeptical about Willie's abilities. He wanted to test his Greek, so he took a copy of Homer with tiny type, and to his amazement, Willie read it easily! At six years and nine months, he was translating Homer and Virgil into English. A year later, his uncle said that William was learning French and Italian so easily that he asked to read Homer in French. He loved the Iliad -- he carried it around with him and quoted his favorite parts. A month after he turned eight, he was with a group of people visiting The Scalp [a rock formation in the Barnaslingan mountains] near Dublin. He was so enthralled with the scenery that he made an announcement about it in Latin. At nine and a half, he wanted to learn Sanskrit. Three months later, he was enthusiastic to learn Oriental languages. At ten and a half, he was studying Arabic and Persian. When he was almost twelve, he wrote a book for publication about "Syriac Grammar" in the Syriac language. When he was fourteen, Mirza Abul Hassan Khan, the Persian [Iranian] ambassador visited Dublin, and William wrote the ambassador a letter in Persian. The English translation is included in Graves' biography. Unfortunately, William's father died around this time. His mother had died two years earlier. Different family members on both sides of the family took care of William and his three sisters.
When William was fifteen, he became interested in science. At first, it was just a way to take a break from his linguistic studies. He wrote in his journal in November, 1820 that he had started reading Isaac Newton's "Principia." He started studying astronomy by observing eclipses, occultations [when one object passes across and hides another], and similar celestial events. At age sixteen he was working with conic sections [defined on Wikipedia] and studying pendulums. He was ill for awhile and moved to Dublin to recover. In May 1822, he was studying differential calculus [a field of calculus related to the rates at which quantities change] and reading Pierre-Simon LaPlace's "Mecanique Celeste" [Celestial Mechanics]. He criticized the part of LaPlace's book about the parallelogram of forces [related to vectors; finding what happens when two forces are applied to an object]. That same year, he wrote his first poems. His varied and erratic interests took on a more definite course of studying when he had to prepare for entrance to the University of Dublin. His tutor, Charles Boyton, was a distinguished scholar -- but he told William that he wouldn't be of much use to him since William was quite capable of tutoring the tutor! Eliza Hamilton [probably his sister, 1807-1851, a poet], who wrote this down, added, 'Yet Charles Boyton promised to be his friend, and said that if he ever saw William starting to become upset by the excitement his presence would draw, and the recognition he would attract, he would be sure to tell him about it.' When he started college, he excelled in every subject, out-distancing the other students. In his first exam, he placed first in Classics, and first in mathematics, and received the Chancellor's prize for a poem on the Ionian Isles, and another award for his poem on Eustace de St. Pierre. [Eustace de St. Pierre begins "The sun has set on Calais' walls, the gloom is deepening thro' her halls." It's included in Robert Perceval Graves' biography, on page 672.]
Many people who knew him said that William had a heart for forming friendships. One of his friends from those days was the writer Maria Edgeworth [she was 37 years older than he was]. She wrote to her sister about, 'young Mr. Hamilton, an admirable Crichton ['the admirable James Crichton' was a brilliant Scottish scholar from the 1500's] who is eighteen years old, a real prodigy of many talents. Dr. Brinkley [probably astronomer John Brinkley, who was then about 60] says he might be a second Isaac Newton, quiet, gentle, and simple.' His sister Eliza, to whom he was affectionately devoted, wrote this to him in 1824:
'I've been imagining you in my mind -- you're in your study at Cumberland Street, with your copy of Xenophon [the Greek philosopher] and your other books on the table. And I can see you, with your transcendent face deep in thought, first sitting down, then walking about, sometimes rubbing your hands together in satisfaction, and sometimes bursting out with a quote from some heroic poem in a foreign language, using your own internal serious voice that can sound like like a ventriloquist when you talk to yourself in your own room -- and even sometimes when you're not actually alone!'
Everyone who knew Hamilton will remember how, even in his later years, he had two distinct voices. One was a high treble, and the other was a deep bass. He would alternate between the two -- not only when he was talking to a friend, but even when he was giving a lecture on some serious subject, such as the mysteries of quaternions [a type of algebraic number] in front of the Royal Irish Academy! Those who knew him got so used to it that they would be surprised when a stranger commented on how ludicrous he sounded.
Hamilton was fortunate in finding a career that was worthy of his talents while he was still so young. He hadn't even finished with his undergraduate studies when he was offered a position in the University of Dublin [Trinity College]. This is how it happened:
In 1826, John Brinkley became the Bishop of Cloyne, so he gave up his position as Professor of Astronomy at Trinity College. Hamilton had attracted such notice that, even though he was still just a 21 year old undergraduate, he was the first person who came to mind as Brinkley's replacement. Actually, he was so multi-talented, that if a vacancy had come up in classics, or mathematics, or metaphysics, or Oriental languages, he probably would have been suggested for any of those! But his friends thought he would be perfect for the astronomy position because of the brilliance he had shown in his mathematical article called "Theory of Systems of Rays" [related to light rays and optics]. This abstract work created a new branch in the field of optics and led to a discovery a few years later that made him famous around the world.
At first, Hamilton felt like it would be presumptuous to take on such an elevated position, so he went away to the country and continued studying to get his degree. Other distinguished candidates came forward, including some from Cambridge and some from Trinity College in Dublin. But Hamilton received a letter from his old tutor, Charles Boyton. Boyton urged him to come back to Dublin because the Board was favorably disposed to consider him. So he agreed, and in June 1827, he was unanimously selected to succeed John Brinkley as the Professor of Astronomy at Trinity College [University of Dublin]. Almost everyone approved -- except for John Brinkley himself. He greatly admired Hamilton's brilliance and talents, but he thought it would have been better for Hamilton to get a Fellowship first. That would have given him more freedom to choose a field of intellectual pursuit. Brinkley thought, with good reason, that a genius like Hamilton would be stifled by the routine and tedious nature of astronomical work. Now that we can look back at Hamilton's life, we can see that he was wrong. It's true that Hamilton never became a skilled astronomical observer, but the seclusion and privacy of the observatory was a perfect environment for him to work at the great projects that his life was devoted to. His work not only distinguished him, but also brought glory to the University of Dublin and the entire country of Ireland.
In his earlier years at Dunsink, Hamilton made some attempts to make practical use of the telescopes, but he didn't really have the aptitude for that kind of work. In fact, exposure to the night air wasn't good for his health, so he gradually focused more on the mathematical research that he had already shown such talent for. Although he earned his greatest fame in pure mathematics, he always claimed, and justly so, that he had a valid claim to the title of astronomer [and not just mathematician]. Later in his life, he commented on this in a striking way. His friend Augustus De Morgan wrote him a letter recommending that he read Robert Grant's book, "History of Physical Astronomy." After he had looked at the book, Hamilton wrote this to his friend:
'The book is valuable, and does credit to its author. But I hope you'll pardon me if I find the title somewhat amusing. He calls his book 'History of Physical Astronomy from the Earliest Ages to the Middle of the Nineteenth Century,' but then fails to even mention any of Sir W. R. Hamilton's discoveries in the theory of the 'Dynamics of the Heavens.'"
The closeness of the two friends accounts for the tone of the letter. He goes on to elaborate his complaint. He writes about how Carl Gustav Jacob Jacobi called him 'the Joseph Louis Lagrange of Ireland' in Manchester in 1842, and William Fishburn Donkin said that, 'The analytical theory of dynamics as it now exists is mostly due to the research of Joseph-Louis LaGrange, Simeon Denis Poisson, Sir William R. Hamilton, and Carl Gustav Jacob Jacobi. Their works on this subject can hardly be paralleled for their elegance and importance in any branch of mathematics.' And then Hamilton hints at successes he had had in clarifying the difficult subject of Planetary Perturbations [disturbances in the orbits of massive bodies]. Even if these had been his only contributions to science, he would have had an illustrious career. But compared with his other intellectual work, these seem relatively insignificant.
Hamilton's most famous achievement during his early years at the observatory was the discovery of conical refraction [has to do with light going through a lens]. This is one of those rare instances where a clever calculation predicted a startling result, and that startling result was confirmed through observation. This made him world famous. Thus, by the time he was twenty-seven, he had already achieved an amount of intellectual success that would have been remarkable for a man of seventy.
Along with his fame came several friendships. There were Thomas Romney Robinson, John Herschel, and others that he corresponded with regularly. Graves' biography includes his correspondence with Coleridge, and letters to his female correspondents, including Maria Edgeworth, Lady Dunraven [Caroline Wyndham-Quin, Countess of Dunraven], and Lady Campbell [Pamela Fitzgerald, wife of Major-General Guy Campbell, Baronet of St. Cross Mede]. Many of their letters are related to literary matters mingled with friendly pleasantries. They show that those who knew him regarded him with affection and respect. There are also letters to his beloved sisters, brimming over with such exaggerated affection that most sisters would smile to imagine their own brothers writing such gushing sentiments. He wrote letters to other young ladies that he admired, although Graves describes his love affairs as often troubled. His attention wasn't always reciprocated, and even the charms of mathematics couldn't ease his disappointment in love. But he finally married Helen Maria Bayly in April 1833. Many years later, he told De Morgan that his marriage was as happy as he expected, and happier than he deserved. There were two sons -- William and Archibald -- and a daughter, Helen, who later married Archdeacon John O'Regan.
Hamilton's most interesting relationship in his early years was his friendship with the poet William Wordsworth. It started when Hamilton visited Keswick [in the Lake District of Cumbria in Cumberland, England] three months after he had become Astronomer Royal [in 1827; Hamilton was 22, Wordsworth was 57]. The first evening they met, something happened that showed that they were intrigued with one another. Hamilton wrote about it in a letter to his sister Eliza:
'Wordsworth walked back with our group as far as their lodge. After we said goodnight to Mrs. Harrison, I offered to walk back with him while the rest of my group went on to the hotel. He accepted, and our conversation was so interesting that by the time we reached his home about a mile away, he suggested walking me part of the way back to Ambleside, and I accepted eagerly. When he turned to go back to his home, I walked him back there again! By the time I finally reached my hotel after all this walking, it was very late.'
Hamilton showed Wordsworth a poem he had written called "It Haunts Me Yet." Here is Wordsworth's response:
'I can assure you with a clear conscience that, in my judgment, your verses are infused with the poetic spirit. They are obviously inspired from strong feeling. I was especially impressed with the sixth and seventh stanzas. They made me teary, and my voice wavered as I read them aloud. And that's all I need to say. Now for the constructive criticism. I'm sure you won't get your feelings hurt if I tell you that the workmanship could be better, but that's to be expected from such a young writer . . .
Everyone in my household says hello very fondly. I have seldom -- in fact, never -- parted from someone after such a short acquaintance and regretted it more. I hope we will meet again.'
They continued to write affectionately after that, and even towards the end of his life, the time he spent in Rydal [Wordsworth's house] was a treasured memory that he mentioned frequently. Wordsworth visited Hamilton at the observatory in 1829, and there's a beautiful shady path that's still called "Wordsworth's Walk."
Hamilton liked to write a sonnet on almost every occasion that he could write a poem about, and he loved to share his poems with all his friends. When William Whewell was writing his Bridgewater Treatises in 1833, he wrote to Hamilton:
'The sonnet you showed me expressed much better than I could the feeling I was aiming for in my book. I was planning to ask your permission to use it in my Preface, but my friends thought I should write my Preface in my own prose, no matter how much better your poem would be.'
[The Bridgewater Treatises were eight treatises commissioned by Francis Egerton, Earl of Bridgewater, that were supposed to explore "the power, wisdom, and goodness of God, as manifested in the creation." Whewell, who was a mathematician, scientist, and Master of Trinity College, wrote his treatise on astronomy and physics.]
The first world-transforming contribution to theoretical dynamics since Isaac Newton's time was Joseph-Louis Lagrange's discovery of the general equations of motion [in 1788; Lagrangian mechanics calculates kinetic and potential energy, where Newton's calculated force]. The next step was William Rowan Hamilton's even more comprehensive method ["Hamilton's principle" is built on Lagrange's work and has to do with stationary action; it can be applied to electromagnetic and gravitational fields, and quantum mechanics]. Hamilton wrote to William Whewell about it in March 1834:
'As far as my recent article that I sent off to London a day or two ago, it's merely mathematical and deductive. I thought about calling it the 'Mecanique Analytique' of Lagrange, 'a scientific poem.' I talked about Dynamics -- or, the science of force -- as being related to 'power acting by law in space and time.' But in every other respect, it's as unpoetical and unmetaphysical as even my most serious friends could wish.'
There is no more beautiful chapter in the history of mathematical philosophy than Hamilton's dynamical theory. It doesn't require any complicated math symbols, and it isn't specific to any single problem. It is an all-embracing theory that gives an understanding of the most appropriate method for applying force to matter. The fact that this theory is so general has somewhat limited some of the ways it can be applied. For example, it isn't as familiar to students of higher mathematics because of the difficulty of writing it into exams: one respected professor complained that Hamilton's essay on dynamics was so abstract that he wasn't able to come up with practical examples in order to test students.
This is part of a letter that Professor Sylvester, an accomplished mathematician, wrote to Hamilton in September 1841. It shows how his theory was appreciated by those who understood it.
'Believe me, sir, one of my greatest regrets about leaving Great Britain is that I won't sometimes run into other masters of my art -- especially you. The acquaintance, or conversation, or even the mere notice of others in my field has the power to inspire. It almost gives fresh life to one's understanding, and the courage and faith that are so necessary to the efforts of invention. I remember the golden moments I spent at your house in Dunsirk. I will probably never experience those kinds of moments again.
Even while I'm so far away and in a less eminent arena, I will be calmly satisfied to observe your blazing course in the grand world of discovery. The national honour you bring to your country may be the only kind of luxury for the rich that can't be bought at the expense of the comforts of the million.'
Hamilton enjoyed studying metaphysics when he wanted a break from mathematics. In 1834, he was studying the philosopher Immanuel Kant. What did this author of "Lectures on Quaternions" and of the essay "Algebra as the Science of Pure Time" think about Kant's "Critique of the Pure Reason?" Here's what he wrote to Viscount Adare [Edwin Richard Wyndham-Quin] in July 1834:
'I read a large part of Kant's 'Critique of the Pure Reason,' and I think it's marvelously clear and very convincing. In spite of some previous preparation from George Berkeley [Berkeley developed the philosophy of subjective idealism; Kant rejected his theory] and from my own thoughts, I think I've learned a lot from Kant's perspective about 'Space and Time.' But most of my enjoyment is because I recognize opinions that I've been familiar with for a long time as I read his works, but they're expressed and combined more clearly and systematically by him . . . I think Kant is more indebted to Berkeley than he admits or even realizes. He mocks him by calling him 'Gutem' [German for 'good'] Berkeley as if he was a good man who meant well and was able to shake the world of human thought to its core and start a revolution in thought that resulted in Kant himself.'
Hamilton was a celebrated figure at British Science Association meetings. In 1835, when the Association had their meeting in Dublin, Hamilton was only thirty years old but already famous enough that even among such a brilliant crowd, he was probably the most acclaimed. There was a banquet at Trinity College during the event, and all the scientists gathered in the University's library. The Earl of Mulgrave [Constantine Henry Phipps], who was Lord Lieutenant of Ireland at the time, knighted Hamilton there, saying, 'I am setting the royal and national mark on you to recognize your distinction, but your own genius and work is what earned you that distinction.'
The banquet was right after the ceremony. Graves's biography says, 'There was even more honour when Professor Whewell returned thanks for the toast of the University of Cambridge. He added the words, 'The thing that comes to my mind at this moment is that 130 years ago, another great man was knighted at Trinity College: Isaac Newton.' There was great applause at this compliment.'
Not long after, he had even more consequential recognition. He wrote this in a letter to Robert Graves in November, 1843:
'This was totally unsolicited, and even a surprise to me, but the Queen [Victoria] expressed her approval of granting me £200 a year from the Civil List [a government fund paid to individuals in appreciation of services rendered to their country] for my work in science. The letters from Sir Robert Peel and the Lord Lieutenant of Ireland [Thomas Philip de Grey] that informed me of this grant were even more satisfying to receive than the money, even though the addition to my income is useful and almost necessary.'
Hamilton seemed to be at the peak of his career -- but that was not so. In fact, everything he had accomplished up to this point was more like practice for the gigantic task he still had ahead of him. William Hamilton is mostly remembered for inventing Quaternion Calculus. [Wikipedia defines quaternions as "a number system that extends the complex numbers" and are "applied to mechanics in three-dimensional space."] Most of his adult life was devoted to this branch of mathematics. He wrote about becoming completely used to the new ways of thinking that came from working with quaternions. In later life, he happened to see an old copy of his famous classic article on Dynamics, so he read it with eager interest. He says he was gratified that he was still able to follow its logic without much trouble, but he felt like it was a work from an outdated era of analysis that had become obsolete.
The revolution that applying quaternion symbols to calculations brought to mathematics can be compared with the advance made by Descartes [Cartesian x and y coordinates in algebra]. This book isn't long enough to fully explain quaternions, but this letter that Hamilton wrote to his son Archibald on his deathbed twenty-two years later tells about how he made the discovery:
'I can still remember the exact month -- October 1843. I was coming home after visiting Cork and Parsonstown [perhaps Lord Rosse's Birr Castle telescope?] for a meeting of the British Science Association. For a long time, these laws of multiplication had been in the back of my mind, but now the desire to discover them became strong and urgent. Sometimes I talked about it with you. Every morning in early October, I would come down for breakfast, and you and your little brother William Edwin would ask, 'Well, papa, can you multiply triplets yet?' And I would sadly shake my head and say, 'No, I can only add and subtract them.'
But on the 16th, which was the Monday of the Council meeting at the Royal Irish Academy, I was walking in that direction to lead the meeting. Your mother and I were walking along the Royal Canal -- maybe she drove and met me there. We chatted together, but all the time, there was an undercurrent going on in my mind, and suddenly it dawned on me. I felt the importance of it immediately. It felt like an electric circuit closed and a spark flash sparked forth, as if to usher in many long years of direct thought and work by me, if I lived long enough, and others if I could pass on the discovery. I could foresee all of this in an instant. As unphilosophical as it sounds, I couldn't resist the impulse to carve the fundamental formula which contains the solution right into a stone of Brougham Bridge. Of course, the inscription wore away a long time ago. But there's a more enduring record on the Academy's Council records. The entry for October 16, 1843 records that I asked for permission to read an article on Quaternions at the next general meeting, which was Monday, November 13.
[Wikipedia's entry on "Quaternion" has a picture of a plaque on Brougham Bridge commemorating the event of Oct 16, 1843.]
He wrote a similar letter describing the event to Professor Peter Guthrie Tait, and another to Rev. John William Stubbs:
'Tomorrow will be the fifteenth birthday of the Quaternions. They burst into life fully grown on October 16, 1843 while I was walking to Dublin with Lady Hamilton, when we came up to Brougham Bridge. My sons have called it Quaternion Bridge ever since! I pulled out a small notebook (which I still have) and made an entry, feeling that this was a project worthy of devoting the next ten or fifteen years of my life. But that might be simply because I felt like a problem that had haunted me for the previous fifteen years had been solved at that moment.
Did anybody before ever have the thought of establishing this kind of system utilizing two geometrically opposed facts -- the fact that two lines or areas in space always give a positive product? I don't think anybody thought of it until I was led to it in October, 1843, while I was trying to extend my old theories of algebraic couples, and of algebra as the science of pure time. As far as geometrical addition of lines as equivalent to composition of motions (and as performed by the same rules), that is an essential part of my theory, but it doesn't only apply to my theory. This view of addition could have come to any of a number of other mathematicians.'
Tourists will continue to visit the spot on the bridge where the theory of Quaternions originated. Maybe while they admire the graceful bridge, they'll wish that someone had refreshed the inscription that he carved into the bridge. But no one ever did, and the inscription is gone forever.
Ten years after the discovery, Hamilton published his great volume, "Lectures on Quaternions" in Dublin. The scientific world acclaimed the book, not only because of Hamilton's reputation, but because of the novelty and importance of his new branch of calculus. His good friend, Sir John Herschel, wrote him a letter in his own typical style:
'I must heartily congratulate you on publishing your book -- I compliment you on finding the words, ore rotundo [rounded], for that seething mass of thought in your head which sometimes sends out sparks, and sometimes smokes and shakes the ground around you, but has now broken forth in a real eruption with a stream of lava and and a shower of enriching ash.
'That's enough of metaphor and simile. It will take any man a whole year to work through your book, and half a lifetime to digest it. I am so glad to see it finished!'
And here is what William Rowan Hamilton wrote to the physicist Humphrey Lloyd:
'I hope I'm becoming more modest about discovering quaternions, but when I see how their principles might be expanded in the future, it seems to me that this discovery is as important to the mid-nineteenth century as Isaac Newton's discovery of fluxions at the end of the seventeenth century.'
When the President of the Royal Irish Academy, Bartholomew Lloyd, had died in 1837, three candidates were suggested to replace him. One was his son, Humphrey Lloyd, and the other two were Hamilton and Archbishop Richard Whately. Humphrey Lloyd wanted Hamilton to have the position, and didn't like his own name being put forward against Hamilton. At the same time, Hamilton wanted to withdraw his name so Humphrey Lloyd would get the position! The Fellows at the college preferred Humphrey Lloyd, since he was more accustomed to a college life than Hamilton, and Lloyd was world-famous for his scientific work with conical refraction research. The vote ended up with Hamilton winning. Lloyd came in second, and Whately was a distant third. It ended happily for all of them, though -- both Lloyd and Whately felt that Hamilton was the better choice, and Hamilton chose both of them to be Vice-Presidents of the Academy. [Lloyd succeeded Hamilton as president in 1846.]
In a previous chapter, I wrote about how John Herschel spent some time at the Cape of Good Hope in order to catalogue the stars of the southern hemisphere to complete the work his father had done in the northern hemisphere. When he returned to England in 1838, there was a banquet in his honour, and Hamilton was the one who was chosen to propose the toast to Herschel. Hamilton remembered that banquet as one of the two times he had been in the company of his very good friend Augustus De Morgan.
Also in 1838, the Royal Irish Academy decided to award medals to scientists who wrote articles of unusual merit. Two scientists were candidates for for the first medal -- Hamilton, for his article on "Algebra, as the Science of Pure Time," and James MacCullagh for his article on the laws of crystalline reflexion and refraction. The medal went to MacCullagh -- partly due to Hamilton's own efforts! Hamilton had gotten a letter from John Herschel talking about the importance of MacCullagh's work in such glowing terms that the award went to him. Hamilton, as chair of the Academy, was the one to award him the medal. The speech he gave on the occasion expressed his own conviction that MacCullagh's work was excellent and deserving of the medal. This circumstance is important to note because, during his entire scientific career, MacCullagh was the only person that Hamilton ever had a disagreement with concerning priority. It happened a few years before this medal when MacCullagh had tried to claim that he was the one who discovered conical refraction. [This maths.tcd.ie page on conical refraction tells a little about Hamilton producing an article with Humphrey Lloyd, presumably in order to establish himself as the discoverer.] Hamilton alludes to that prior incident in a letter to the Marquis of Northampton from June 1838:
'Although some previous circumstances prevented me from calling MacCullagh a friend, I was pleased to be able to do justice to his intellectual stature. I think he was not only gratified, but also touched. Maybe in the future, he'll regard me with the same friendly feelings that I want to harbor towards him.'
Sometimes Hamilton would start to keep a journal, but he never kept up with any of them. That's unfortunate for a person trying to write his biography, but he made up for it by keeping copies of all the letters he sent as well as other keepsakes. In fact, he would take exact notes of the most trivial things on scraps of paper in an almost amusing way. He often made a note of the individual who carried his letter to the post office and the exact time it was despatched. He also kept the letters he received, all jumbled up with his scientific manuscripts. He had papers all over his study, and even in other parts of his house. If he laid aside a letter, within a few hours, it was lost within his mass of papers, although he said they would sometimes come 'eddying' to the surface later while looking for something else.
The large collection of "Lectures on Quaternions" had been published in 1853, and Hamilton had been justly honoured for completing such a great immortal work -- but how was the printer's bill going to be paid? Printing such a thick book wasn't cheap. It didn't seem likely that all the copies would sell, but even if they did, they would barely earn enough to pay for the expense of publishing. Where was the money going to come from? The Board of Trinity College had already contributed £200, but that was still £100 short. Even William Rowan Hamilton himself, the man who discovered the quaternions, was stressed and anxious about this. But his faithful friend Humphrey Lloyd was now a member of the Board, and used his influence to take care of the rest of the expense. I should also mention that, even with his pension and his regular salary, Hamilton always seemed to be struggling financially. He wrote to De Morgan, 'Although I'm not embarrassingly poor, I'm not rich, either.' In spite of his world-wide fame for his discoveries, the only financial royalties he saw from his works was from the Icosian Game, a mathematical puzzle he invented in 1856. A friend of his urged an enterprising publisher to buy the copyright for £25 and sell the game commercially. But the public just wasn't interested, and it didn't make money for the publisher.
After his great book on Quaternions was published, Hamilton relaxed a bit and indulged in even more reading and writing. He corresponded regularly with his friend, the Irish poet Aubrey de Vere, and he had many other friends that he kept up with through letters. His sister, the poetess Eliza Mary Hamilton, died in 1851, and that greatly affected him. She left him all of her papers to either preserve or destroy, but it was only after four years that he could overcome his grief enough to open her box of personal letters.
We can see Hamilton's religious side in his correspondence, especially with Aubrey de Vere, who had converted to Roman Catholicism in 1851. [Hamilton was deeply religious, but Protestant.] In August 1855, Hamilton wrote:
'It seems evident to both of us that under these circumstances, no matter how much we wish it could be different, there cannot be the same degree of intimacy between us as before regarding the nature of things, or of minds. We can no longer talk with the same degree of openness on every subject that occurs to us. Out of basic courtesy, we'll both have to be on our guard so as not to say something that might offend or even cause pain to the other. And yet we were once so close. But I hope we still retain the same regard, esteem, and appreciation for each other. Our friendship is built on so many associations from my early youth and your boyhood [de Vere was nine years younger than Hamilton] that neither of us can forget those memories. We have grown so close that two or three respectable friendships could be carved out of the fragments of our former and never-forgotten intimacy. I don't exaggerate when I quote the words, "Heu, quanto minus est cum reliquis versari quam tui meminisse!" [This is a quote by William Shenstone that translates: 'Oh, how disappointing it is to associate with those who remain, when I remember you!"]
In 1858, Hamilton began corresponding with physicist Peter Guthrie Tait about Quaternions. Hamilton was pleased that such a skilled mathematician wanted to become familiar with this new kind of calculus. Tait later wrote "Elementary Treatise on Quaternions" with Hamilton's advice, although it wasn't published until after Hamilton had died. Anyone who wants an introduction to the subject would probably prefer Tait's book, since Hamilton's is much longer.
In 1861, Quaternions began to draw attention outside of England. It attracted the notice of the German mathematician August Ferdinand Mobius, author of "Der Barycentrische Calcul" [Calculus of the Centres of Gravity]. His work had some relationship with Quaternions. Hamilton enjoyed hearing how his work was noticed by such eminent mathematicians, and the recognition motivated him to become even more absorbed in his work. During the last few years of his life, he became a bit of a recluse. His focus seemed to grow sharper so that he was able to spend longer periods of time in long, continuous study, and his leisure time became more brief and less frequent.
Sometimes he worked twelve hours at a stretch. He would look up from his work to put out his candles after a late night of fascinating research, and notice the sun coming up! A regular routine was impossible when he was in the middle of what he called one of his 'mathematical trances.' Sometimes he would snatch a few hours of sleep and grab a meal in the intervals of sober periods in between his Quaternion attacks. If he got hungry, he would see if there was anything quick to snack on readily at hand. When he was thirsty, he would visit the locker [tavern?]. The only potential blemish on his character is that he might have made these visits a bit too often. [One biographer suggested that Hamilton had a drinking problem and an unhappy marriage, but a close look at his life and character doesn't show that at all. Anne van Weerden and Steven Wepster wrote an article about that for the BSHM Bulletin: Journal of the British Society for the History of Mathematics.]
He sometimes took a rare break from his all-absorbing study of Quaternions. One time he was curious about the exact date of the Hegira [when Muhammed and his followers left Mecca and emigrated to Medina], so he calculated back and arrived at July 15, 622 A.D. He was pleased when he later discovered that Herschel had calculated and arrived at the same date! He continued to enjoy reading and reflecting on metaphysics, and corresponding with friends. He wrote a long letter to Clement Mansfield Ingleby about Ingleby's book "Introduction to Metaphysics." In his letter, Hamilton mentioned a flaw in his own eyesight. He had an issue with his eyes working together, so he got used to seeing the world as two distinct images at the same time. He commented that using a stereoscope [a 3-D viewer; a View-Master] was helpful to his vision -- he had never realized before then that the two separate images he had always seen were supposed to be blended into a single image! He wondered if this was related to the phenomena of binocular vision [animals with binocular vision can see in 3-D], and speculated in his letter that perhaps there's no need for binocular vision to correctly assess distance since, as he wrote, 'I am quite sure that I'm able to see distance with each eye separately.'
Hamilton began 1865 as diligent as ever. He was corresponding with Arthur Cayley and George Salmon. In April 1865, he wrote to a friend that his health hadn't been very good for the past few years, and overwork had taken its toll. In addition, stress after receiving another printer's bill for the publication of his book 'Elements of Quaternions' hadn't done much for his peace of mind. In fact, it caused him serious anxiety until the day he died. Ultimately, the entire cost -- about £500 -- was paid by the College, like the previous bill had been. It had always been feared that the book wouldn't sell well. After it went out of print, a single used copy was sold for as much as £5! [Converted to US dollars and inflation from 1865 to 2016, it works out to $94.]
William Hamilton visited Dublin for the last time in May, 1865. A few days later, he had a severe attack of gout [related to arthritis; symptoms are joint pain and fever], and then became alarming ill and had epileptic convulsions. But he rallied a little, and was working on his book "Elements of Quaternions" by the end of June. One happy incident brightened his final days. The National Academy of Science had just formed in America. They wanted some foreign associates from around the world, and when they discussed who those associates should be, Hamilton's name was first on the list. He was informed that he was being awarded this great distinction by a two-thirds majority vote.
He was still working on the Table of Contents for his "Elements" book in August. He wrote a letter to Benjamin Gould in America, thanking him for the honour of being made a member of the National Academy of Science. On September 2, he called Robert Graves to come and see him at the observatory, and told him that he sensed that the end was approaching. He said that Psalm 145 expressed his thoughts and feeling very accurately, and that he wanted to affirm his faith and thankfulness by taking communion. He died that very afternoon. He was sixty years old. He was buried in Mount Jerome Cemetery.
Many people wrote letters and publicly expressed their sadness at his death. Sir John Herschel wrote this to Hamilton's widow, Helen:
'Of all the scientific friends I have been deprived of, there is none for whom I grieve more -- not only because of his splendid talents, but also for his excellent disposition and perfectly simple manners that were so great, yet free from pretensions.'
Augustus De Morgan, his old mathematical companion, also wrote to Lady Hamilton:
'I have called him one of my dearest friends, and I meant it. We corresponded intimately for more than twenty-five years of friendly agreement and even disagreement, and always had the most cordial interest in each other. And yet we barely knew what each other looked like. I met him once around 1830 at Charles Babbage's breakfast table. That was the only time we we conversed in person in our entire lives. At a dinner given in honour of Sir John Herschel around 1838 when Herschel had just come back from the Cape in South Africa, I saw Hamilton from a distance, but it was too crowded for us to speak to one another. And that is all I ever saw in this world of the man whose friendly letters were one of my most cherished social pleasures and greatest intellectual delights.'
Augustus De Morgan wrote a memoir called "Sir W. R. Hamilton" for the "Gentleman's Magazine and Historical Review" in 1866 [you can read it online at the Trinity College Dublin website]. He wrote an excellent description of his friend's character embellished with personal memories and episodes. He alluded to the quaint confusion of papers in his study, among other things. But, as chaotic as his papers looked to an outsider, there was a sort of order in the mass that was only known to Hamilton himself. Any time a servant tried to tidy them up, Hamilton would be thrown into "a good honest thundering passion."
Two brilliant mathematicians couldn't have been more different than Hamilton and De Morgan. Hamilton had a highly poetical temperament, and De Morgan was a practical realist. One time Hamilton sent some sonnets to his friend, and De Morgan responded by giving him advice about how to write a will! Hamilton often filled his letters with metaphysical subtleties, but De Morgan preferred battles of wit over such topics of logic as quantifying the predicate. He was exquisitely witty, and although Hamilton enjoyed his jokes, he hardly ever joked himself. In fact, his rare attempts at humour went over like a lead balloon. And yet these two scientists were perfectly in tune with each other. Hamilton's work on Quaternions and Dynamics, his literary tastes, his metaphysics, and his poetry were sincerely welcomed by De Morgan. De Morgan's letters always expressed the kindliest interest in Hamilton's concerns, and Hamilton always responded wholeheartedly to De Morgan's letters.
We hope that a collection of Hamilton's works will be published soon in honour of his memory. It would be a credit to his University and a benefit for science. It would be wonderful to have a collection to showcase his early work in optical theory, his later work that caused him to be considered the Joseph-Louis Lagrange of Ireland, and his creation of Quaternion Calculus that has given new capabilities to the human intellect.
[You can read some of William Rowan's Hamilton's poetry online at The Net Advance of Physics RETRO.]
by Sir Robert S. Ball, 1895 (paraphrased by Leslie Noelani Laurio)
To view the table of contents for the rest of this book, click here.
Sir William Rowan Hamilton, 1805-1865
Integrated Newtonian dynamics and invented a new branch of calculus related to Quaternions
William Rowan Hamilton was born right at midnight between August 3 and 4, 1805, in Dublin. His father, Archibald Hamilton, was a solicitor [a type of lawyer], and William was the fourth of his nine children. William's ancestors were mostly Irish gentlemen -- but his grandmother on his mother's side of the family was Scottish. When he was still a baby, his parents decided to let his uncle, James Hamilton, who was a clergyman in Trim, in County Meath [Ireland] educate him since he ran a school. James's sister, Sidney, lived with him, and William spent his childhood there.
In Robert Perceval Graves' book "Life of Sir William Rowan Hamilton" [three volumes on line at archive.org: 1, 2, 3], there are some letters included in which Aunt Sidney wrote to William's mother in Dublin to let her know how William was doing. They are extraordinary. When he was three years old, his aunt wrote that William was a "hopeful blade" because he was a strong little boy, which she illustrated by relating that William was able to make boys older than himself run away from him! A month later, she wrote that William was brought in to class to read the Bible out loud in front of the first graders to shame them for not reading as well as William. At three years old! His Uncle James worked hard to teach him, but his aunt said she couldn't imagine how he was able to pick up so much of what he taught because "he never stops playing and jumping about!" When he was 4 years and three months old, he had dinner with the vicar and amused everyone there by reading a book for them no matter whether the book was turned upside down, or sideways! Aunt Sidney wrote to his mother that "Willie is a most sensible little creature, but at the same time has a great deal of roguery." When he was four and five months, he went to town to visit his mother, and his mother wrote this to her sister:
'His ability to recite is astonishing. His obvious and accurate knowledge of geography is unbelievable. He can even sketch out the countries on paper with a pencil and cut them out! They're not exact, but close enough that anyone who knew the countries would recognize them. But that's nothing compared with how well he reads Latin, Greek, and Hebrew.'
Aunt Sidney wrote that as soon as Willie got back to Trim, he was eager to continue with his lessons. He wouldn't even eat his breakfast until his uncle had heard him recite his Hebrew and explained to him the importance of proper pronunciation. At age five, he went with his uncle to visit a friend and recited long passages from John Dryden. Another gentleman was there who was skeptical about Willie's abilities. He wanted to test his Greek, so he took a copy of Homer with tiny type, and to his amazement, Willie read it easily! At six years and nine months, he was translating Homer and Virgil into English. A year later, his uncle said that William was learning French and Italian so easily that he asked to read Homer in French. He loved the Iliad -- he carried it around with him and quoted his favorite parts. A month after he turned eight, he was with a group of people visiting The Scalp [a rock formation in the Barnaslingan mountains] near Dublin. He was so enthralled with the scenery that he made an announcement about it in Latin. At nine and a half, he wanted to learn Sanskrit. Three months later, he was enthusiastic to learn Oriental languages. At ten and a half, he was studying Arabic and Persian. When he was almost twelve, he wrote a book for publication about "Syriac Grammar" in the Syriac language. When he was fourteen, Mirza Abul Hassan Khan, the Persian [Iranian] ambassador visited Dublin, and William wrote the ambassador a letter in Persian. The English translation is included in Graves' biography. Unfortunately, William's father died around this time. His mother had died two years earlier. Different family members on both sides of the family took care of William and his three sisters.
When William was fifteen, he became interested in science. At first, it was just a way to take a break from his linguistic studies. He wrote in his journal in November, 1820 that he had started reading Isaac Newton's "Principia." He started studying astronomy by observing eclipses, occultations [when one object passes across and hides another], and similar celestial events. At age sixteen he was working with conic sections [defined on Wikipedia] and studying pendulums. He was ill for awhile and moved to Dublin to recover. In May 1822, he was studying differential calculus [a field of calculus related to the rates at which quantities change] and reading Pierre-Simon LaPlace's "Mecanique Celeste" [Celestial Mechanics]. He criticized the part of LaPlace's book about the parallelogram of forces [related to vectors; finding what happens when two forces are applied to an object]. That same year, he wrote his first poems. His varied and erratic interests took on a more definite course of studying when he had to prepare for entrance to the University of Dublin. His tutor, Charles Boyton, was a distinguished scholar -- but he told William that he wouldn't be of much use to him since William was quite capable of tutoring the tutor! Eliza Hamilton [probably his sister, 1807-1851, a poet], who wrote this down, added, 'Yet Charles Boyton promised to be his friend, and said that if he ever saw William starting to become upset by the excitement his presence would draw, and the recognition he would attract, he would be sure to tell him about it.' When he started college, he excelled in every subject, out-distancing the other students. In his first exam, he placed first in Classics, and first in mathematics, and received the Chancellor's prize for a poem on the Ionian Isles, and another award for his poem on Eustace de St. Pierre. [Eustace de St. Pierre begins "The sun has set on Calais' walls, the gloom is deepening thro' her halls." It's included in Robert Perceval Graves' biography, on page 672.]
Many people who knew him said that William had a heart for forming friendships. One of his friends from those days was the writer Maria Edgeworth [she was 37 years older than he was]. She wrote to her sister about, 'young Mr. Hamilton, an admirable Crichton ['the admirable James Crichton' was a brilliant Scottish scholar from the 1500's] who is eighteen years old, a real prodigy of many talents. Dr. Brinkley [probably astronomer John Brinkley, who was then about 60] says he might be a second Isaac Newton, quiet, gentle, and simple.' His sister Eliza, to whom he was affectionately devoted, wrote this to him in 1824:
'I've been imagining you in my mind -- you're in your study at Cumberland Street, with your copy of Xenophon [the Greek philosopher] and your other books on the table. And I can see you, with your transcendent face deep in thought, first sitting down, then walking about, sometimes rubbing your hands together in satisfaction, and sometimes bursting out with a quote from some heroic poem in a foreign language, using your own internal serious voice that can sound like like a ventriloquist when you talk to yourself in your own room -- and even sometimes when you're not actually alone!'
Everyone who knew Hamilton will remember how, even in his later years, he had two distinct voices. One was a high treble, and the other was a deep bass. He would alternate between the two -- not only when he was talking to a friend, but even when he was giving a lecture on some serious subject, such as the mysteries of quaternions [a type of algebraic number] in front of the Royal Irish Academy! Those who knew him got so used to it that they would be surprised when a stranger commented on how ludicrous he sounded.
Hamilton was fortunate in finding a career that was worthy of his talents while he was still so young. He hadn't even finished with his undergraduate studies when he was offered a position in the University of Dublin [Trinity College]. This is how it happened:
In 1826, John Brinkley became the Bishop of Cloyne, so he gave up his position as Professor of Astronomy at Trinity College. Hamilton had attracted such notice that, even though he was still just a 21 year old undergraduate, he was the first person who came to mind as Brinkley's replacement. Actually, he was so multi-talented, that if a vacancy had come up in classics, or mathematics, or metaphysics, or Oriental languages, he probably would have been suggested for any of those! But his friends thought he would be perfect for the astronomy position because of the brilliance he had shown in his mathematical article called "Theory of Systems of Rays" [related to light rays and optics]. This abstract work created a new branch in the field of optics and led to a discovery a few years later that made him famous around the world.
At first, Hamilton felt like it would be presumptuous to take on such an elevated position, so he went away to the country and continued studying to get his degree. Other distinguished candidates came forward, including some from Cambridge and some from Trinity College in Dublin. But Hamilton received a letter from his old tutor, Charles Boyton. Boyton urged him to come back to Dublin because the Board was favorably disposed to consider him. So he agreed, and in June 1827, he was unanimously selected to succeed John Brinkley as the Professor of Astronomy at Trinity College [University of Dublin]. Almost everyone approved -- except for John Brinkley himself. He greatly admired Hamilton's brilliance and talents, but he thought it would have been better for Hamilton to get a Fellowship first. That would have given him more freedom to choose a field of intellectual pursuit. Brinkley thought, with good reason, that a genius like Hamilton would be stifled by the routine and tedious nature of astronomical work. Now that we can look back at Hamilton's life, we can see that he was wrong. It's true that Hamilton never became a skilled astronomical observer, but the seclusion and privacy of the observatory was a perfect environment for him to work at the great projects that his life was devoted to. His work not only distinguished him, but also brought glory to the University of Dublin and the entire country of Ireland.
In his earlier years at Dunsink, Hamilton made some attempts to make practical use of the telescopes, but he didn't really have the aptitude for that kind of work. In fact, exposure to the night air wasn't good for his health, so he gradually focused more on the mathematical research that he had already shown such talent for. Although he earned his greatest fame in pure mathematics, he always claimed, and justly so, that he had a valid claim to the title of astronomer [and not just mathematician]. Later in his life, he commented on this in a striking way. His friend Augustus De Morgan wrote him a letter recommending that he read Robert Grant's book, "History of Physical Astronomy." After he had looked at the book, Hamilton wrote this to his friend:
'The book is valuable, and does credit to its author. But I hope you'll pardon me if I find the title somewhat amusing. He calls his book 'History of Physical Astronomy from the Earliest Ages to the Middle of the Nineteenth Century,' but then fails to even mention any of Sir W. R. Hamilton's discoveries in the theory of the 'Dynamics of the Heavens.'"
The closeness of the two friends accounts for the tone of the letter. He goes on to elaborate his complaint. He writes about how Carl Gustav Jacob Jacobi called him 'the Joseph Louis Lagrange of Ireland' in Manchester in 1842, and William Fishburn Donkin said that, 'The analytical theory of dynamics as it now exists is mostly due to the research of Joseph-Louis LaGrange, Simeon Denis Poisson, Sir William R. Hamilton, and Carl Gustav Jacob Jacobi. Their works on this subject can hardly be paralleled for their elegance and importance in any branch of mathematics.' And then Hamilton hints at successes he had had in clarifying the difficult subject of Planetary Perturbations [disturbances in the orbits of massive bodies]. Even if these had been his only contributions to science, he would have had an illustrious career. But compared with his other intellectual work, these seem relatively insignificant.
Hamilton's most famous achievement during his early years at the observatory was the discovery of conical refraction [has to do with light going through a lens]. This is one of those rare instances where a clever calculation predicted a startling result, and that startling result was confirmed through observation. This made him world famous. Thus, by the time he was twenty-seven, he had already achieved an amount of intellectual success that would have been remarkable for a man of seventy.
Along with his fame came several friendships. There were Thomas Romney Robinson, John Herschel, and others that he corresponded with regularly. Graves' biography includes his correspondence with Coleridge, and letters to his female correspondents, including Maria Edgeworth, Lady Dunraven [Caroline Wyndham-Quin, Countess of Dunraven], and Lady Campbell [Pamela Fitzgerald, wife of Major-General Guy Campbell, Baronet of St. Cross Mede]. Many of their letters are related to literary matters mingled with friendly pleasantries. They show that those who knew him regarded him with affection and respect. There are also letters to his beloved sisters, brimming over with such exaggerated affection that most sisters would smile to imagine their own brothers writing such gushing sentiments. He wrote letters to other young ladies that he admired, although Graves describes his love affairs as often troubled. His attention wasn't always reciprocated, and even the charms of mathematics couldn't ease his disappointment in love. But he finally married Helen Maria Bayly in April 1833. Many years later, he told De Morgan that his marriage was as happy as he expected, and happier than he deserved. There were two sons -- William and Archibald -- and a daughter, Helen, who later married Archdeacon John O'Regan.
Hamilton's most interesting relationship in his early years was his friendship with the poet William Wordsworth. It started when Hamilton visited Keswick [in the Lake District of Cumbria in Cumberland, England] three months after he had become Astronomer Royal [in 1827; Hamilton was 22, Wordsworth was 57]. The first evening they met, something happened that showed that they were intrigued with one another. Hamilton wrote about it in a letter to his sister Eliza:
'Wordsworth walked back with our group as far as their lodge. After we said goodnight to Mrs. Harrison, I offered to walk back with him while the rest of my group went on to the hotel. He accepted, and our conversation was so interesting that by the time we reached his home about a mile away, he suggested walking me part of the way back to Ambleside, and I accepted eagerly. When he turned to go back to his home, I walked him back there again! By the time I finally reached my hotel after all this walking, it was very late.'
Hamilton showed Wordsworth a poem he had written called "It Haunts Me Yet." Here is Wordsworth's response:
'I can assure you with a clear conscience that, in my judgment, your verses are infused with the poetic spirit. They are obviously inspired from strong feeling. I was especially impressed with the sixth and seventh stanzas. They made me teary, and my voice wavered as I read them aloud. And that's all I need to say. Now for the constructive criticism. I'm sure you won't get your feelings hurt if I tell you that the workmanship could be better, but that's to be expected from such a young writer . . .
Everyone in my household says hello very fondly. I have seldom -- in fact, never -- parted from someone after such a short acquaintance and regretted it more. I hope we will meet again.'
They continued to write affectionately after that, and even towards the end of his life, the time he spent in Rydal [Wordsworth's house] was a treasured memory that he mentioned frequently. Wordsworth visited Hamilton at the observatory in 1829, and there's a beautiful shady path that's still called "Wordsworth's Walk."
Hamilton liked to write a sonnet on almost every occasion that he could write a poem about, and he loved to share his poems with all his friends. When William Whewell was writing his Bridgewater Treatises in 1833, he wrote to Hamilton:
'The sonnet you showed me expressed much better than I could the feeling I was aiming for in my book. I was planning to ask your permission to use it in my Preface, but my friends thought I should write my Preface in my own prose, no matter how much better your poem would be.'
[The Bridgewater Treatises were eight treatises commissioned by Francis Egerton, Earl of Bridgewater, that were supposed to explore "the power, wisdom, and goodness of God, as manifested in the creation." Whewell, who was a mathematician, scientist, and Master of Trinity College, wrote his treatise on astronomy and physics.]
The first world-transforming contribution to theoretical dynamics since Isaac Newton's time was Joseph-Louis Lagrange's discovery of the general equations of motion [in 1788; Lagrangian mechanics calculates kinetic and potential energy, where Newton's calculated force]. The next step was William Rowan Hamilton's even more comprehensive method ["Hamilton's principle" is built on Lagrange's work and has to do with stationary action; it can be applied to electromagnetic and gravitational fields, and quantum mechanics]. Hamilton wrote to William Whewell about it in March 1834:
'As far as my recent article that I sent off to London a day or two ago, it's merely mathematical and deductive. I thought about calling it the 'Mecanique Analytique' of Lagrange, 'a scientific poem.' I talked about Dynamics -- or, the science of force -- as being related to 'power acting by law in space and time.' But in every other respect, it's as unpoetical and unmetaphysical as even my most serious friends could wish.'
There is no more beautiful chapter in the history of mathematical philosophy than Hamilton's dynamical theory. It doesn't require any complicated math symbols, and it isn't specific to any single problem. It is an all-embracing theory that gives an understanding of the most appropriate method for applying force to matter. The fact that this theory is so general has somewhat limited some of the ways it can be applied. For example, it isn't as familiar to students of higher mathematics because of the difficulty of writing it into exams: one respected professor complained that Hamilton's essay on dynamics was so abstract that he wasn't able to come up with practical examples in order to test students.
This is part of a letter that Professor Sylvester, an accomplished mathematician, wrote to Hamilton in September 1841. It shows how his theory was appreciated by those who understood it.
'Believe me, sir, one of my greatest regrets about leaving Great Britain is that I won't sometimes run into other masters of my art -- especially you. The acquaintance, or conversation, or even the mere notice of others in my field has the power to inspire. It almost gives fresh life to one's understanding, and the courage and faith that are so necessary to the efforts of invention. I remember the golden moments I spent at your house in Dunsirk. I will probably never experience those kinds of moments again.
Even while I'm so far away and in a less eminent arena, I will be calmly satisfied to observe your blazing course in the grand world of discovery. The national honour you bring to your country may be the only kind of luxury for the rich that can't be bought at the expense of the comforts of the million.'
Hamilton enjoyed studying metaphysics when he wanted a break from mathematics. In 1834, he was studying the philosopher Immanuel Kant. What did this author of "Lectures on Quaternions" and of the essay "Algebra as the Science of Pure Time" think about Kant's "Critique of the Pure Reason?" Here's what he wrote to Viscount Adare [Edwin Richard Wyndham-Quin] in July 1834:
'I read a large part of Kant's 'Critique of the Pure Reason,' and I think it's marvelously clear and very convincing. In spite of some previous preparation from George Berkeley [Berkeley developed the philosophy of subjective idealism; Kant rejected his theory] and from my own thoughts, I think I've learned a lot from Kant's perspective about 'Space and Time.' But most of my enjoyment is because I recognize opinions that I've been familiar with for a long time as I read his works, but they're expressed and combined more clearly and systematically by him . . . I think Kant is more indebted to Berkeley than he admits or even realizes. He mocks him by calling him 'Gutem' [German for 'good'] Berkeley as if he was a good man who meant well and was able to shake the world of human thought to its core and start a revolution in thought that resulted in Kant himself.'
Hamilton was a celebrated figure at British Science Association meetings. In 1835, when the Association had their meeting in Dublin, Hamilton was only thirty years old but already famous enough that even among such a brilliant crowd, he was probably the most acclaimed. There was a banquet at Trinity College during the event, and all the scientists gathered in the University's library. The Earl of Mulgrave [Constantine Henry Phipps], who was Lord Lieutenant of Ireland at the time, knighted Hamilton there, saying, 'I am setting the royal and national mark on you to recognize your distinction, but your own genius and work is what earned you that distinction.'
The banquet was right after the ceremony. Graves's biography says, 'There was even more honour when Professor Whewell returned thanks for the toast of the University of Cambridge. He added the words, 'The thing that comes to my mind at this moment is that 130 years ago, another great man was knighted at Trinity College: Isaac Newton.' There was great applause at this compliment.'
Not long after, he had even more consequential recognition. He wrote this in a letter to Robert Graves in November, 1843:
'This was totally unsolicited, and even a surprise to me, but the Queen [Victoria] expressed her approval of granting me £200 a year from the Civil List [a government fund paid to individuals in appreciation of services rendered to their country] for my work in science. The letters from Sir Robert Peel and the Lord Lieutenant of Ireland [Thomas Philip de Grey] that informed me of this grant were even more satisfying to receive than the money, even though the addition to my income is useful and almost necessary.'
Hamilton seemed to be at the peak of his career -- but that was not so. In fact, everything he had accomplished up to this point was more like practice for the gigantic task he still had ahead of him. William Hamilton is mostly remembered for inventing Quaternion Calculus. [Wikipedia defines quaternions as "a number system that extends the complex numbers" and are "applied to mechanics in three-dimensional space."] Most of his adult life was devoted to this branch of mathematics. He wrote about becoming completely used to the new ways of thinking that came from working with quaternions. In later life, he happened to see an old copy of his famous classic article on Dynamics, so he read it with eager interest. He says he was gratified that he was still able to follow its logic without much trouble, but he felt like it was a work from an outdated era of analysis that had become obsolete.
The revolution that applying quaternion symbols to calculations brought to mathematics can be compared with the advance made by Descartes [Cartesian x and y coordinates in algebra]. This book isn't long enough to fully explain quaternions, but this letter that Hamilton wrote to his son Archibald on his deathbed twenty-two years later tells about how he made the discovery:
'I can still remember the exact month -- October 1843. I was coming home after visiting Cork and Parsonstown [perhaps Lord Rosse's Birr Castle telescope?] for a meeting of the British Science Association. For a long time, these laws of multiplication had been in the back of my mind, but now the desire to discover them became strong and urgent. Sometimes I talked about it with you. Every morning in early October, I would come down for breakfast, and you and your little brother William Edwin would ask, 'Well, papa, can you multiply triplets yet?' And I would sadly shake my head and say, 'No, I can only add and subtract them.'
But on the 16th, which was the Monday of the Council meeting at the Royal Irish Academy, I was walking in that direction to lead the meeting. Your mother and I were walking along the Royal Canal -- maybe she drove and met me there. We chatted together, but all the time, there was an undercurrent going on in my mind, and suddenly it dawned on me. I felt the importance of it immediately. It felt like an electric circuit closed and a spark flash sparked forth, as if to usher in many long years of direct thought and work by me, if I lived long enough, and others if I could pass on the discovery. I could foresee all of this in an instant. As unphilosophical as it sounds, I couldn't resist the impulse to carve the fundamental formula which contains the solution right into a stone of Brougham Bridge. Of course, the inscription wore away a long time ago. But there's a more enduring record on the Academy's Council records. The entry for October 16, 1843 records that I asked for permission to read an article on Quaternions at the next general meeting, which was Monday, November 13.
[Wikipedia's entry on "Quaternion" has a picture of a plaque on Brougham Bridge commemorating the event of Oct 16, 1843.]
He wrote a similar letter describing the event to Professor Peter Guthrie Tait, and another to Rev. John William Stubbs:
'Tomorrow will be the fifteenth birthday of the Quaternions. They burst into life fully grown on October 16, 1843 while I was walking to Dublin with Lady Hamilton, when we came up to Brougham Bridge. My sons have called it Quaternion Bridge ever since! I pulled out a small notebook (which I still have) and made an entry, feeling that this was a project worthy of devoting the next ten or fifteen years of my life. But that might be simply because I felt like a problem that had haunted me for the previous fifteen years had been solved at that moment.
Did anybody before ever have the thought of establishing this kind of system utilizing two geometrically opposed facts -- the fact that two lines or areas in space always give a positive product? I don't think anybody thought of it until I was led to it in October, 1843, while I was trying to extend my old theories of algebraic couples, and of algebra as the science of pure time. As far as geometrical addition of lines as equivalent to composition of motions (and as performed by the same rules), that is an essential part of my theory, but it doesn't only apply to my theory. This view of addition could have come to any of a number of other mathematicians.'
Tourists will continue to visit the spot on the bridge where the theory of Quaternions originated. Maybe while they admire the graceful bridge, they'll wish that someone had refreshed the inscription that he carved into the bridge. But no one ever did, and the inscription is gone forever.
Ten years after the discovery, Hamilton published his great volume, "Lectures on Quaternions" in Dublin. The scientific world acclaimed the book, not only because of Hamilton's reputation, but because of the novelty and importance of his new branch of calculus. His good friend, Sir John Herschel, wrote him a letter in his own typical style:
'I must heartily congratulate you on publishing your book -- I compliment you on finding the words, ore rotundo [rounded], for that seething mass of thought in your head which sometimes sends out sparks, and sometimes smokes and shakes the ground around you, but has now broken forth in a real eruption with a stream of lava and and a shower of enriching ash.
'That's enough of metaphor and simile. It will take any man a whole year to work through your book, and half a lifetime to digest it. I am so glad to see it finished!'
And here is what William Rowan Hamilton wrote to the physicist Humphrey Lloyd:
'I hope I'm becoming more modest about discovering quaternions, but when I see how their principles might be expanded in the future, it seems to me that this discovery is as important to the mid-nineteenth century as Isaac Newton's discovery of fluxions at the end of the seventeenth century.'
When the President of the Royal Irish Academy, Bartholomew Lloyd, had died in 1837, three candidates were suggested to replace him. One was his son, Humphrey Lloyd, and the other two were Hamilton and Archbishop Richard Whately. Humphrey Lloyd wanted Hamilton to have the position, and didn't like his own name being put forward against Hamilton. At the same time, Hamilton wanted to withdraw his name so Humphrey Lloyd would get the position! The Fellows at the college preferred Humphrey Lloyd, since he was more accustomed to a college life than Hamilton, and Lloyd was world-famous for his scientific work with conical refraction research. The vote ended up with Hamilton winning. Lloyd came in second, and Whately was a distant third. It ended happily for all of them, though -- both Lloyd and Whately felt that Hamilton was the better choice, and Hamilton chose both of them to be Vice-Presidents of the Academy. [Lloyd succeeded Hamilton as president in 1846.]
In a previous chapter, I wrote about how John Herschel spent some time at the Cape of Good Hope in order to catalogue the stars of the southern hemisphere to complete the work his father had done in the northern hemisphere. When he returned to England in 1838, there was a banquet in his honour, and Hamilton was the one who was chosen to propose the toast to Herschel. Hamilton remembered that banquet as one of the two times he had been in the company of his very good friend Augustus De Morgan.
Also in 1838, the Royal Irish Academy decided to award medals to scientists who wrote articles of unusual merit. Two scientists were candidates for for the first medal -- Hamilton, for his article on "Algebra, as the Science of Pure Time," and James MacCullagh for his article on the laws of crystalline reflexion and refraction. The medal went to MacCullagh -- partly due to Hamilton's own efforts! Hamilton had gotten a letter from John Herschel talking about the importance of MacCullagh's work in such glowing terms that the award went to him. Hamilton, as chair of the Academy, was the one to award him the medal. The speech he gave on the occasion expressed his own conviction that MacCullagh's work was excellent and deserving of the medal. This circumstance is important to note because, during his entire scientific career, MacCullagh was the only person that Hamilton ever had a disagreement with concerning priority. It happened a few years before this medal when MacCullagh had tried to claim that he was the one who discovered conical refraction. [This maths.tcd.ie page on conical refraction tells a little about Hamilton producing an article with Humphrey Lloyd, presumably in order to establish himself as the discoverer.] Hamilton alludes to that prior incident in a letter to the Marquis of Northampton from June 1838:
'Although some previous circumstances prevented me from calling MacCullagh a friend, I was pleased to be able to do justice to his intellectual stature. I think he was not only gratified, but also touched. Maybe in the future, he'll regard me with the same friendly feelings that I want to harbor towards him.'
Sometimes Hamilton would start to keep a journal, but he never kept up with any of them. That's unfortunate for a person trying to write his biography, but he made up for it by keeping copies of all the letters he sent as well as other keepsakes. In fact, he would take exact notes of the most trivial things on scraps of paper in an almost amusing way. He often made a note of the individual who carried his letter to the post office and the exact time it was despatched. He also kept the letters he received, all jumbled up with his scientific manuscripts. He had papers all over his study, and even in other parts of his house. If he laid aside a letter, within a few hours, it was lost within his mass of papers, although he said they would sometimes come 'eddying' to the surface later while looking for something else.
The large collection of "Lectures on Quaternions" had been published in 1853, and Hamilton had been justly honoured for completing such a great immortal work -- but how was the printer's bill going to be paid? Printing such a thick book wasn't cheap. It didn't seem likely that all the copies would sell, but even if they did, they would barely earn enough to pay for the expense of publishing. Where was the money going to come from? The Board of Trinity College had already contributed £200, but that was still £100 short. Even William Rowan Hamilton himself, the man who discovered the quaternions, was stressed and anxious about this. But his faithful friend Humphrey Lloyd was now a member of the Board, and used his influence to take care of the rest of the expense. I should also mention that, even with his pension and his regular salary, Hamilton always seemed to be struggling financially. He wrote to De Morgan, 'Although I'm not embarrassingly poor, I'm not rich, either.' In spite of his world-wide fame for his discoveries, the only financial royalties he saw from his works was from the Icosian Game, a mathematical puzzle he invented in 1856. A friend of his urged an enterprising publisher to buy the copyright for £25 and sell the game commercially. But the public just wasn't interested, and it didn't make money for the publisher.
After his great book on Quaternions was published, Hamilton relaxed a bit and indulged in even more reading and writing. He corresponded regularly with his friend, the Irish poet Aubrey de Vere, and he had many other friends that he kept up with through letters. His sister, the poetess Eliza Mary Hamilton, died in 1851, and that greatly affected him. She left him all of her papers to either preserve or destroy, but it was only after four years that he could overcome his grief enough to open her box of personal letters.
We can see Hamilton's religious side in his correspondence, especially with Aubrey de Vere, who had converted to Roman Catholicism in 1851. [Hamilton was deeply religious, but Protestant.] In August 1855, Hamilton wrote:
'It seems evident to both of us that under these circumstances, no matter how much we wish it could be different, there cannot be the same degree of intimacy between us as before regarding the nature of things, or of minds. We can no longer talk with the same degree of openness on every subject that occurs to us. Out of basic courtesy, we'll both have to be on our guard so as not to say something that might offend or even cause pain to the other. And yet we were once so close. But I hope we still retain the same regard, esteem, and appreciation for each other. Our friendship is built on so many associations from my early youth and your boyhood [de Vere was nine years younger than Hamilton] that neither of us can forget those memories. We have grown so close that two or three respectable friendships could be carved out of the fragments of our former and never-forgotten intimacy. I don't exaggerate when I quote the words, "Heu, quanto minus est cum reliquis versari quam tui meminisse!" [This is a quote by William Shenstone that translates: 'Oh, how disappointing it is to associate with those who remain, when I remember you!"]
In 1858, Hamilton began corresponding with physicist Peter Guthrie Tait about Quaternions. Hamilton was pleased that such a skilled mathematician wanted to become familiar with this new kind of calculus. Tait later wrote "Elementary Treatise on Quaternions" with Hamilton's advice, although it wasn't published until after Hamilton had died. Anyone who wants an introduction to the subject would probably prefer Tait's book, since Hamilton's is much longer.
In 1861, Quaternions began to draw attention outside of England. It attracted the notice of the German mathematician August Ferdinand Mobius, author of "Der Barycentrische Calcul" [Calculus of the Centres of Gravity]. His work had some relationship with Quaternions. Hamilton enjoyed hearing how his work was noticed by such eminent mathematicians, and the recognition motivated him to become even more absorbed in his work. During the last few years of his life, he became a bit of a recluse. His focus seemed to grow sharper so that he was able to spend longer periods of time in long, continuous study, and his leisure time became more brief and less frequent.
Sometimes he worked twelve hours at a stretch. He would look up from his work to put out his candles after a late night of fascinating research, and notice the sun coming up! A regular routine was impossible when he was in the middle of what he called one of his 'mathematical trances.' Sometimes he would snatch a few hours of sleep and grab a meal in the intervals of sober periods in between his Quaternion attacks. If he got hungry, he would see if there was anything quick to snack on readily at hand. When he was thirsty, he would visit the locker [tavern?]. The only potential blemish on his character is that he might have made these visits a bit too often. [One biographer suggested that Hamilton had a drinking problem and an unhappy marriage, but a close look at his life and character doesn't show that at all. Anne van Weerden and Steven Wepster wrote an article about that for the BSHM Bulletin: Journal of the British Society for the History of Mathematics.]
He sometimes took a rare break from his all-absorbing study of Quaternions. One time he was curious about the exact date of the Hegira [when Muhammed and his followers left Mecca and emigrated to Medina], so he calculated back and arrived at July 15, 622 A.D. He was pleased when he later discovered that Herschel had calculated and arrived at the same date! He continued to enjoy reading and reflecting on metaphysics, and corresponding with friends. He wrote a long letter to Clement Mansfield Ingleby about Ingleby's book "Introduction to Metaphysics." In his letter, Hamilton mentioned a flaw in his own eyesight. He had an issue with his eyes working together, so he got used to seeing the world as two distinct images at the same time. He commented that using a stereoscope [a 3-D viewer; a View-Master] was helpful to his vision -- he had never realized before then that the two separate images he had always seen were supposed to be blended into a single image! He wondered if this was related to the phenomena of binocular vision [animals with binocular vision can see in 3-D], and speculated in his letter that perhaps there's no need for binocular vision to correctly assess distance since, as he wrote, 'I am quite sure that I'm able to see distance with each eye separately.'
Hamilton began 1865 as diligent as ever. He was corresponding with Arthur Cayley and George Salmon. In April 1865, he wrote to a friend that his health hadn't been very good for the past few years, and overwork had taken its toll. In addition, stress after receiving another printer's bill for the publication of his book 'Elements of Quaternions' hadn't done much for his peace of mind. In fact, it caused him serious anxiety until the day he died. Ultimately, the entire cost -- about £500 -- was paid by the College, like the previous bill had been. It had always been feared that the book wouldn't sell well. After it went out of print, a single used copy was sold for as much as £5! [Converted to US dollars and inflation from 1865 to 2016, it works out to $94.]
William Hamilton visited Dublin for the last time in May, 1865. A few days later, he had a severe attack of gout [related to arthritis; symptoms are joint pain and fever], and then became alarming ill and had epileptic convulsions. But he rallied a little, and was working on his book "Elements of Quaternions" by the end of June. One happy incident brightened his final days. The National Academy of Science had just formed in America. They wanted some foreign associates from around the world, and when they discussed who those associates should be, Hamilton's name was first on the list. He was informed that he was being awarded this great distinction by a two-thirds majority vote.
He was still working on the Table of Contents for his "Elements" book in August. He wrote a letter to Benjamin Gould in America, thanking him for the honour of being made a member of the National Academy of Science. On September 2, he called Robert Graves to come and see him at the observatory, and told him that he sensed that the end was approaching. He said that Psalm 145 expressed his thoughts and feeling very accurately, and that he wanted to affirm his faith and thankfulness by taking communion. He died that very afternoon. He was sixty years old. He was buried in Mount Jerome Cemetery.
Many people wrote letters and publicly expressed their sadness at his death. Sir John Herschel wrote this to Hamilton's widow, Helen:
'Of all the scientific friends I have been deprived of, there is none for whom I grieve more -- not only because of his splendid talents, but also for his excellent disposition and perfectly simple manners that were so great, yet free from pretensions.'
Augustus De Morgan, his old mathematical companion, also wrote to Lady Hamilton:
'I have called him one of my dearest friends, and I meant it. We corresponded intimately for more than twenty-five years of friendly agreement and even disagreement, and always had the most cordial interest in each other. And yet we barely knew what each other looked like. I met him once around 1830 at Charles Babbage's breakfast table. That was the only time we we conversed in person in our entire lives. At a dinner given in honour of Sir John Herschel around 1838 when Herschel had just come back from the Cape in South Africa, I saw Hamilton from a distance, but it was too crowded for us to speak to one another. And that is all I ever saw in this world of the man whose friendly letters were one of my most cherished social pleasures and greatest intellectual delights.'
Augustus De Morgan wrote a memoir called "Sir W. R. Hamilton" for the "Gentleman's Magazine and Historical Review" in 1866 [you can read it online at the Trinity College Dublin website]. He wrote an excellent description of his friend's character embellished with personal memories and episodes. He alluded to the quaint confusion of papers in his study, among other things. But, as chaotic as his papers looked to an outsider, there was a sort of order in the mass that was only known to Hamilton himself. Any time a servant tried to tidy them up, Hamilton would be thrown into "a good honest thundering passion."
Two brilliant mathematicians couldn't have been more different than Hamilton and De Morgan. Hamilton had a highly poetical temperament, and De Morgan was a practical realist. One time Hamilton sent some sonnets to his friend, and De Morgan responded by giving him advice about how to write a will! Hamilton often filled his letters with metaphysical subtleties, but De Morgan preferred battles of wit over such topics of logic as quantifying the predicate. He was exquisitely witty, and although Hamilton enjoyed his jokes, he hardly ever joked himself. In fact, his rare attempts at humour went over like a lead balloon. And yet these two scientists were perfectly in tune with each other. Hamilton's work on Quaternions and Dynamics, his literary tastes, his metaphysics, and his poetry were sincerely welcomed by De Morgan. De Morgan's letters always expressed the kindliest interest in Hamilton's concerns, and Hamilton always responded wholeheartedly to De Morgan's letters.
We hope that a collection of Hamilton's works will be published soon in honour of his memory. It would be a credit to his University and a benefit for science. It would be wonderful to have a collection to showcase his early work in optical theory, his later work that caused him to be considered the Joseph-Louis Lagrange of Ireland, and his creation of Quaternion Calculus that has given new capabilities to the human intellect.
[You can read some of William Rowan's Hamilton's poetry online at The Net Advance of Physics RETRO.]
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