**Robert Murphy**was the son of the shoemaker John Murphy and his wife Margaret. Although we do not know his exact date of birth, we know that he was baptized in the Church of Ireland on 8 March 1807 which would suggest that he was born near the end of 1806. Robert was, according to some sources, the third of his parents' seven children and, according to other sources, the sixth child of seven boys and two girls. He was brought up in the family home on Beecher Street, Mallow. John Murphy clearly had a reasonable education since he served as the parish clerk. Tragedy struck the family, however, in January 1814 when John Murphy died. Robert was only seven years old at this time and the death of his father meant that from that time on the family were in extreme poverty. His life became even more difficult when he was eleven years old for at that time he was run over by a cart outside his home and his thighbone was fractured in the accident. He spent a year in bed recovering from this and even when he was able to leave his bed after a year he still had to spend a further six months walking with great difficulty.

It is often the case that misfortune has its positive side and this appears to be true for Murphy for it was during the time that he was confined to bed that he discovered that he had a talent for mathematics. His mother could not afford to buy him mathematics books but she found an old Cork almanac containing problems on navigation and positional astronomy solved using trigonometry. He spent many a happy hour in bed with this book and, now that his mathematical talent was clear, he was given a copy of Euclid's *Elements* and an algebra textbook.

John Mulcahy, a graduate of Trinity College Dublin in 1830, was appointed to the Chair of Natural Philosophy at Queen's College, Galway before the university opened in 1849, but before taking up the appointment was transferred to the Chair of Mathematics. It is not John Mulcahy but rather it is his father, Mulcahy Senior, who played a vital role in Robert Murphy's life. Mulcahy Senior was a teacher in Cork and he posed mathematical problems in a local paper. He received some remarkable solutions from someone who signed themselves "Mallow" and, intrigued to find out who the mathematical genius was, travelled to Mallow. There he learnt to his amazement that the solutions were from Robert Murphy, a child on crutches of around 12 years of age. Mulcahy Senior spoke to Murphy's mother and immediately understood that there was no way that the family could afford to educate their son. He then made strenuous efforts to arrange for Murphy to get an education.

Mulcahy spoke to one of Murphy's neighbours, a Mr Croker, saying:-

He persuaded Croker to help finance Murphy's education and he also spoke to Mr Hopley, the headmaster of the local school, who agreed to cover Murphy's fees and the cost of books at his school. Murphy spent three years between 1819 and 1823 at Mr Hopley's school where he was taught by a Mr Armstrong. In 1823 Murphy, supported by Mulcahy, Hopley and Croker, applied to enter Trinity College, Dublin. As part of his application he attached examples of his mathematical work. His application, however, was rejected. With only three years formal education and presenting mathematical work which those viewing his application failed to understand, led to this outcome. The reader will notice that we have here a not too dissimilar situation to that faced by the young Évariste Galois five years later. Such setbacks were not going to stop Murphy doing mathematics.Mr Croker, you have a second Sir Isaac Newton in Mallow: pray look after him.

The Reverend John Mackey had published a pamphlet entitled *A Method of Making a Cube a Double of a Cube, Founded on the Principles of Elementary Geometry*. We know now, of course, that doubling the cube using ruler and compass is impossible but this was not known in the 1820s. We know little about John Mackey other than he was originally from Cashel and, at the time he wrote his pamphlet, he was at Maynooth College. Murphy had amused himself by studying the classical Greek problems and believed that they were insoluble. He quickly found the error in John Mackey ruler and compass construction of the cube root of 2 and in 1824 published his own pamphlet entitled *'Refutation of a Pamphlet Written by the Rev. John Mackey Entitled "A Method of Making a Cube a Double of a Cube, Founded on the Principles of Elementary Geometry," wherein His Principles Are Proved Erroneous and the Required Solution Not Yet Obtained*. Murphy began his account by looking briefly at the history of duplication the cube and trisecting the angle. He then wrote:-

Note the confidence of the eighteen year old boy with little formal education. Murphy then goes on to correctly point out the errors in the Reverend John Mackey's arguments using only Euclid'sBut amongst all the attempts which have been made for the solution of the duplication, there has not been one more foolish or more erroneous, than that of the Rev John Mackey; which being masked under the appearance of truth, consists of a collection of false propositions.

*Elements*and Cardan's

*Ars Magna*(more precisely the Cardan-Tartaglia formula for solving cubic equations). Murphy ends his 19-page pamphlet by writing:-

Now Murphy had a stroke of luck. McCarthy was a junior fellow of Gonville and Caius College, Cambridge, who lived in Cork and when he returned home for a vacation he met Murphy. Collecting together some of Murphy's mathematics he returned to Cambridge and showed them to Robert Woodhouse. Woodhouse glanced at Murphy's work and at first did not rate it highly but, on a second reading, he saw its potential. He wrote to McCarthy saying that he would accept Murphy as a student at Gonville and Caius College, Cambridge, for a fee of £60 if his sponsors could raise that amount. He assured them that this would cover all of Murphy's expenses for the duration of his undergraduate course. Mulcahy, with the help of several friends in Mallow, was able to put together the £60 and Murphy was admitted as a pensioner to Gonville and Caius College, Cambridge, on 7 July 1825.We shall conclude with hoping that Mr Mackey's next attempt will be more successful.

Murphy won the 1st Mathematics Prize in 1826 and went on to graduate with a B.A. in 1829. He was 3^{rd} wrangler, meaning that he was ranked third among the students receiving a first class degree. This excellent performance, quite remarkable for someone with Murphy's minimal formal education before beginning his undergraduate studies, led to Murphy being awarded a Perse Fellowship. At the same time, to help his financial position, he was appointed as Librarian. He gave Hebrew lectures in 1830 (giving six lectures and receiving £10 as payment), and was appointed as a junior dean in charge of discipline and chapel services in October 1831, a position he held until 1833. He was ordained a deacon on 4 June 1831 and gave Greek lectures in 1832. On top of all this he was writing innovative mathematical papers and was commissioned in 1830 to write a book on the mathematical theory of electricity for the use of students at Cambridge. It appeared that the young man had finally made the transition to a top class mathematician. However, his newly found status and his good income was something he could not cope with and the young man became addicted to betting.

Already as an undergraduate there is considerable evidence that Murphy could not handle his finances. In December 1827 he was "advanced thirty pounds out of the Perse and forty pounds out of the College Fund". In December of the following year, at the meeting of the Senior Fellows when Murphy's degree was confirmed, it was "agreed to advance Mr Murphy a loan of fifty pounds". There is another factor in Murphy's problems for his mother, who had given him great moral support during his youth, died in 1832. He was already gambling, making many bets with William Haughton Stokes (1802-1884) and others, and getting himself into severe debt problems. After the death of his mother these problems got worse and he was informed in December 1832 that his position as junior dean was to end. Also the same meeting of the Senior Fellows in December 1832 it was agreed that "sixty pounds be allowed to Mr Murphy for the present year". William Haughton Stokes, who we mentioned above, was the elder brother of the mathematician George Gabriel Stokes, and like Murphy was Irish. William Haughton Stokes had been 16^{th} Wrangler in 1828 and was appointed a fellow of Gonville and Caius College, Cambridge. In the two months, October and November 1832, ten Common Room wagers by Murphy are recorded in the College Wager Book and, in 1833, 21 of the total of 60 bets recorded involve Murphy.

What sort of bets did Murphy make? Many are on political topics such as, "Mr Stokes bets Mr Murphy that there will be at least forty-five members returned from Parliament at the next election." Many are quite strange, for example "Mr Murphy bets Mr Stokes that Napoleon is said to have sacrificed the life of his wife rather than that of his son," or "Mr Paget bets Mr Murphy that Abraham took to himself a wife after the death of Sarah." We note that Mr Paget is George Edward Paget who became a distinguished physician. Many of the bets involved bottles of wine and there is a suggestion that Murphy had problems with alcohol as well as with betting. Although Murphy lost his position as a junior dean in charge of discipline in 1833, he continued to hold other posts at Gonville and Caius College, being a Hebrew lecturer in 1833, 1834 and 1835. He was also a university examiner in mathematics in 1833. However he continued to run up debts throughout Cambridge and by 1836 he was forced to leave the city. The money from his fellowship was withheld from him and used by the College to repay his debts. He had spent the years 1830 to 1836 producing top quality mathematics and we look now at some of the work he published during these years.

Murphy published *On the General Properties of Definite Integrals* (1830), *On the Resolution of Algebraical Equations* (1831), *First Memoir on the Inverse Method of Definite Integrals, with Physical Applications* (1830) and *On Elimination between an Indefinite Number of Unknown Quantities* (1832) in the *Transactions of the Cambridge Philosophical Society*. We can already see the two main areas on which Murphy was working, namely on integral equations and algebraic equations. His next publication was a contribution to mathematical physics, namely the book *Elementary Principles of the Theories of Electricity, Heat, and Molecular Actions. Part I on Electricity *which was designed for use by Cambridge undergraduates. You can read Murphy's Introduction to this book at THIS LINK.

William Thomson writes in *Electricity and Magnetism* (1872):-

Although Green'sIt appears highly probable that[Murphy]may never have had access to Green's 'Essay' at all, and that this is the explanation of the fact(of which any other explanation is scarcely conceivable), that in his 'Treatise on Electricity' he makes no allusion whatever to Green's discoveries, and gives a theory in no respect pushed beyond what had been done by Poisson ...

*Essay*was published in March 1828, it was only seen at this time by those around Nottingham. It is unlikely that Murphy would know of it until Green became an undergraduate at Caius College in October 1833. Murphy's book was certainly completed by June 1833 (this is the date on the Introduction) so it is completely understandable that Green's ideas did not get included in Murphy's book.

On the theory of equations Murphy wrote papers such as *On the Existence of Real or Imaginary Root to Any Equation* (1833), *Further Development of the Existence of a Real or Imaginary Root to Any Equation* (1833), *Analysis of the Roots of Equations* (1837) and the results from these papers were all contained in Murphy's 1839 book *A Treatise on the Theory of Algebraical Equations*. The book also included a description of the work on others on the theory of equations. You can read Murphy's Introduction to this book at THIS LINK.

Perhaps Murphy's most remarkable paper was *First Memoir on the Theory of Analytic Operations* published in 1837 in the *Philosophical Transactions of the Royal Society of London*. In this paper Murphy looks at linear operators on functions. Looking at examples of such operators was not new but Murphy looked at these operators as an algebraic system. He defined addition and multiplication of the linear operators (multiplication being composition). He noted that in general the multiplication was not commutative. Although, of course, he does not use modern terminology, what he had defined was a non-commutative ring. He went on the define the kernel of an operator (which he called the appendage) and showed that an operator was invertible if its kernel was 0. He noted the, now standard, result that if *f*, *g* are invertible operators

*f*.

*g*)

^{-1}=

*g*

^{-1}.

*f*

^{-1}.

*f*

^{-1}.

*g*.

*f*, Lie brackets

*f*.

*g*-

*g*.

*f*, and gives the binomial theorem for non-commuting variables. All of this was eight years before Hamilton's non-commutative system of quaternions was published.

Let us continue to describe the events of Murphy's life. He had been elected to the Royal Society on 5 June 1834 and, in the same year, given a Frankland junior fellowship. After he had to leave Cambridge, he returned briefly to Ireland but went to London in October 1836 since he had learnt that the professor of mathematics at University College, London, George James Pelly White, had drowned, together with his wife, mother and two boatmen, in a pleasure boat accident off Guernsey on 5 October. George Pelly White, a graduate of Trinity College, Cambridge, was only 25 years old when he drowned. Murphy's hope for this chair were not realised since, even before he reached London, De Morgan had been appointed to fill the vacant chair. In fact De Morgan had previously held the chair up to the summer of 1831 when he had resigned in protest over the dismissal of the Professor of Anatomy.

De Morgan, who fully understood Murphy's genius, tried to help him over the next few years. Murphy lived cheaply in London and made some money tutoring privately. Encouraged by De Morgan he also wrote articles for the Society for the Diffusion of Useful Knowledge and for the *Penny Cyclopaedia*. It was while living in London in these difficult circumstances that he wrote the paper on what are now called non-commutative rings which we described above. He still had hopes of returning to Cambridge and he went there in 1838 when he was elected to a Stokes Fellowship by Caius College. Unfortunately this did not provide him with the income he had hoped for since it was confiscated to pay off debts which he still had in Cambridge. Caius College made advance payment so that he could pay off the "Porter, Bedmaker, Laundress, Shoecleaner and Milkman". He was, however, appointed as an examiner of mathematics and natural philosophy at the University of London in October 1838. He continued to hope for a Senior Fellowship at Cambridge but it was not to be.

By this time his health was deteriorating and he had symptoms of lung disease. He battled on, however, continuing to undertake research and to write it up for publication. His book *A Treatise on the Theory of Algebraical Equations* was published in 1839 and provided much need funds. His final works were *Remark on Primitive Radices* (1841), *Calculations of Logarithms by Means of Algebraic Fractions* (1841), and *On Atmospheric Refraction* (1842). As his lung disease got progressively worse his pupils deserted him. Today Murphy is little known but, had he been more able to organise his life, his health may have been better and he may have had the time to produce results of lasting importance of which his genius showed him to be capable. He was buried in an unmarked grave in Kensal Green Cemetery, London.

**Article by:** *J J O'Connor* and *E F Robertson*

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