# Robert Murphy: Mathematician and Physicist

Author(s):
Anthony J. Del Latto (Columbia University) and Salvatore J. Petrilli, Jr. (Adelphi University)

### Introduction

Despite his short life, Robert Murphy (1806-1843) was a mathematician and physicist who “had a true genius for mathematical invention,” according to Augustus De Morgan (1806-1871) [Venn 2009]. Murphy’s mathematical research can be categorized into three areas: Algebraic Equations, Integral Equations, and Operator Calculus [Allaire 2002]. The majority of the scholarship on Murphy is centered around his contributions to physics; however, historians such as Petrova [1978] and Bradley and Allaire [2002] have explored Murphy’s linear operator theory. The purpose of this paper is to provide a unified exposition in which we synthesize and expand on existing accounts of Murphy’s life and mathematical contributions. Additionally, we give an overview of his mathematical papers and accomplishments in hopes of inspiring historians to examine and analyze his original works.

The authors have also provided a detailed bibliography, which lists where all but one of Murphy’s books and papers are available for download, either Google Books or the Journal STORage database (JSTOR). Murphy’s first paper,

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 [1824],

has not been available in the United States. As a result, the authors have provided a transcription of this paper with commentary as an appendix available here.

# Robert Murphy: Mathematician and Physicist - Introduction

Author(s):
Anthony J. Del Latto (Columbia University) and Salvatore J. Petrilli, Jr. (Adelphi University)

### Introduction

Despite his short life, Robert Murphy (1806-1843) was a mathematician and physicist who “had a true genius for mathematical invention,” according to Augustus De Morgan (1806-1871) [Venn 2009]. Murphy’s mathematical research can be categorized into three areas: Algebraic Equations, Integral Equations, and Operator Calculus [Allaire 2002]. The majority of the scholarship on Murphy is centered around his contributions to physics; however, historians such as Petrova [1978] and Bradley and Allaire [2002] have explored Murphy’s linear operator theory. The purpose of this paper is to provide a unified exposition in which we synthesize and expand on existing accounts of Murphy’s life and mathematical contributions. Additionally, we give an overview of his mathematical papers and accomplishments in hopes of inspiring historians to examine and analyze his original works.

The authors have also provided a detailed bibliography, which lists where all but one of Murphy’s books and papers are available for download, either Google Books or the Journal STORage database (JSTOR). Murphy’s first paper,

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 [1824],

has not been available in the United States. As a result, the authors have provided a transcription of this paper with commentary as an appendix available here.

# Robert Murphy: Mathematician and Physicist - The Early Years

Author(s):
Anthony J. Del Latto (Columbia University) and Salvatore J. Petrilli, Jr. (Adelphi University)

### The Early Years

We are fortunate to have an entry on Robert Murphy in the Dictionary of National Biography. This entry, by Thompson Cooper [1894], led the authors to original sources on Murphy, such as a letter written by Augustus De Morgan (1806-1871) and an obituary. Unless otherwise noted, the information in this section (The Early Years) and the following section (Careers in Academia) is from Cooper [1894].

Figure 1. Main street of Mallow, County Cork, Ireland, in 2007 (Source: Photograph by Alison, 2007. Licensed under Creative Commons Attribution-Share Alike 3.0 Unported license.)

##### Life Experiences

Robert Murphy was born in 1806 in Mallow, Cork County, Ireland. We do not know Murphy’s exact birth date, but he was baptized in the Church of Ireland on March 8, 1807 [Barry 1999]. He was the third son (of seven children) of John Murphy, a shoemaker, and Margaret Murphy. (Creedon [2001] claimed that Murphy was the sixth of nine children, but no other references support this claim.) When he was eleven, Murphy was run over by a cart in an accident that resulted in a fractured thighbone. This incident left him bedridden for one year. During this time, Murphy read the works of Euclid (ca. 300 BCE) and studied algebra.

Murphy demonstrated his mathematical ability at a young age when he anonymously responded with ingenious solutions to mathematical problems posed in a local newspaper by a teacher in Cork named Mulcahy. Murphy would end his responses by signing “Mallow” [Barry 1999]. Mulcahy was shocked when he learned that it was a thirteen-year-old boy who was submitting solutions to his problems [Creedon 2001]. Mr. Croker, one of Murphy’s neighbors, provided an account in which Mulcahy said in amazement, “Mr. Croker, you have a second Sir Isaac Newton in Mallow: pray look after him” [Long 1846, pp. 337-338]. Mulcahy then convinced Mr. Hopley, principal of a local school, to pay for the fees and books required for Murphy to attend. After completing his studies at the school, Mulcahy and Hopley sponsored Murphy’s application to Trinity College, Dublin, in 1823. Murphy was not admitted, most likely because of his lack of formal education [Creedon 2001]. Although Murphy’s first attempt to gain entrance into college was unsuccessful, he gained recognition less than a year later by writing a paper that dealt with the three classical Greek problems.

##### Murphy’s Refutation

When he was eighteen, Murphy was recognized for his aforementioned publication,

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 [1824].

Murphy’s pamphlet was cited by the well-known mathematician Augustus De Morgan in his “Budget of Paradoxes” [1864], published in the The Athenæum: Journal of Literature, Science, and the Fine Arts. De Morgan wrote [1864, p. 181]:

This refutation was the production of an Irish boy of eighteen years old, self-educated in mathematics, the son of a shoemaker at Mallow.

Murphy’s [1824] Refutation is something of an enigma, because we know very little about John Mackey. However, it is clear that Mackey thought he had found a way to use a straightedge and compass to double the cube and trisect an angle. With fervor, Murphy set out to demonstrate that Mackey was incorrect [Murphy 1824, pp. iv-v]:

But 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.

Murphy began the Refutation by providing a brief history of these two ancient problems. Additionally, Murphy stated that all he needed was Euclid’s Elements [Elrington 1822] and the “Method” of Girolamo Cardano (1501-1576) to show that Mackey was wrong. Murphy used ideas from classical Euclidean constructions and the algebraic ideas behind the solution to the cubic equation in his demonstrations. After he proved Mackey wrong, Murphy ended his paper with the sly comment, “We shall conclude with hoping that Mr. Mackey’s next attempt will be more successful” [Murphy 1824, p. 19].

The authors have provided a transcription of Murphy's Refutation, with commentary, as an appendix available here.

# Robert Murphy: Mathematician and Physicist - Careers in Academia

Author(s):
Anthony J. Del Latto (Columbia University) and Salvatore J. Petrilli, Jr. (Adelphi University)

Coincidentally, Murphy was presented with another opportunity to gain admission to a university when he met Mr. McCarthy, a junior fellow at Gonville and Caius College, Cambridge, while he was on vacation in Cork [Creedon 2001]. Impressed, Mr. McCarthy brought some of Murphy’s work back to Gonville and Caius and presented it to Professor Robert Woodhouse (1773-1827). Woodhouse saw potential in Murphy’s work and offered him admission to Cambridge as a result [Creedon 2001]. Several people in Mallow, led by Mulcahy, raised the tuition required to send Murphy to Gonville and Caius in October 1825 [Barry 1986]. Unfortunately, little is known about Murphy’s undergraduate studies at Cambridge; however, Venn [2009] reported that Murphy received a “1st. math prize” in 1826. He graduated in 1829, earning a B.A. with a rank of third wrangler. A student was awarded the title of wrangler if he gained first-class honors on the Mathematical Tripos [Barry 1999]. Murphy scored the third highest on the Mathematical Tripos examination.

Figure 2. Murphy’s entry in the Gonville and Caius Matriculation Book, which was made when he arrived in 1825. (Source: Permission granted by the Master and Fellows of Gonville and Caius College, Cambridge.) The entry reads:

Murphy Robert, son of John of Mallow, Ireland, born there: educated at Mallow for three years under Mr. Armstrong; age 18; admitted on 7 July as a pensioner of this College, tutor within the same, and released by death.

Kate McQuillian, Archivist at Gonville and Caius College, provided the above translation of the excerpt from the Gonville and Caius College Matriculation Book. (See Figure 2.) Ms. McQuillian stated that the abrupt ending of the excerpt means that Murphy remained a member of the College until his death.

##### Teacher, Researcher, Deacon, Shoemaker

After graduating, Murphy was appointed a fellow of Gonville and Caius College. This fellowship gave him a small salary, and the opportunity to teach classes – he held a position as a Hebrew and Greek lecturer [Venn 2009] – and tutor students for additional pay. On June 4, 1831, he was ordained a deacon (Chichester) [Venn 2009]. Shortly after, Murphy was elected dean in May 1831. According to Barry [1986], this position paid him handsomely. As dean, Murphy ensured that the rules of the church were enforced. During this time, Murphy began publishing papers in physics and mathematics, which we discuss in the next three sections.

Figure 3. Gonville and Caius College, Cambridge University, England, in 2011 (Source: Public domain)

Murphy received his M.A. in 1832 [Venn 2009]; however, due to a lifestyle that consisted largely of drinking and gambling, he left Cambridge in 1832 when creditors seized his property. He then traveled to West Cork, Ireland, and became a shoemaker; however, his heavy drinking continued [Barry 1986]. Murphy returned to Mallow in 1835 and, with the aid of his friends, he stopped drinking [Barry 1986]. In 1836 he was ready to return to England and restart his career in academia.

While Murphy was in Ireland, he wrote several papers on physics. He earned royalties from his book Elementary Principles of the Theories of Electricity, Heat and Molecular Action: On Electricity [1833], which aided him in paying off a large portion of his debts. As a result of his numerous publications and growing reputation, Murphy resumed his life as a teacher and writer when he moved to London in 1836. He was granted a position as examiner of mathematics and natural philosophy at the University of London in October 1838, which he held until his death.

The next three sections are devoted to Murphy’s contributions to mathematics. Allaire [2002] determined that Murphy’s works fall into three categories: Algebraic Equations, Integral Equations, and Operator Calculus. We adopt these classifications in this paper.

# Robert Murphy: Mathematician and Physicist - Murphy’s Works: Algebraic Equations

Author(s):
Anthony J. Del Latto (Columbia University) and Salvatore J. Petrilli, Jr. (Adelphi University)

### Murphy’s Works: Algebraic Equations

##### Introduction and Methodology

The following works fall within Murphy’s “Algebraic Equations” research area:

• “On the Resolution of Algebraical Equations” [1831],
• “On Elimination between an Indefinite Number of Unknown Quantities” [1832],
• “On the Existence of Real or Imaginary Root to Any Equation” [1833],
• “On the Real Functions of Imaginary Quantities” [1833],
• “Further Development of the Existence of a Real or Imaginary Root to Any Equation” [1833],
• “On the Resolution of Equations in Finite Differences” [1835],
• “Analysis of the Roots of Equations” [1837],
• “Remark on an Error of Fourier in his ‘Analyse des équations’” [1837],
• A Treatise on the Theory of Algebraical Equations [1839],
• “Remark on Primitive Radices” [1841],
• “Calculations of Logarithms by Means of Algebraic Fractions” [1841].

We will discuss only Murphy’s [1839] work, A Treatise on the Theory of Algebraical Equations, because, based on its preface, it is reasonable to conjecture that this book includes a majority of Murphy’s shorter works on this topic. We will first examine Murphy’s treatment of pure analysis because we discovered many of Lagrange’s and Cauchy’s ideas in this section of his work. Then, we will provide a brief overview of his examination of algebraic equations, where we found many notable problems and an example of an original contribution in which he corrected an error made by Joseph Fourier (1768-1830).  Murphy’s book is formatted in a manner that reminds us of modern textbooks in that he presented theory and provided many examples intended to “impress the reader with the steps of the reasoning” [1839, p. iii]. His goal was to provide a unified source on the theory of algebraic equations. A majority of his work was devoted to the study of “rational and integer functions of $$x$$,” by which he meant a polynomial function with integer coefficients having rational roots. However, he noted that a complete source could not be created unless the theory of pure analysis was incorporated within the work. Therefore, he began with a discussion of pure analysis.

Figure 4. Portraits of, left to right, Jean Baptiste Joseph Fourier, Augustin-Louis Cauchy, and Joseph-Louis Lagrange (Sources: Wikimedia Commons (Fourier) and Convergence Portrait Gallery)

##### The Combined Ideas of Lagrange and Cauchy

Within the subject of pure analysis, Murphy wanted to examine topics such as maxima and minima; however, he needed a foundation of calculus to begin with. Murphy adopted Joseph-Louis Lagrange’s (1736-1813) algebraic analysis from Lagrange’s 1797 work Théorie des fonctions analytiques. We will not go into detail regarding Lagrange’s method; however, readers interested in this method can see [Lagrange 1797, pp. 1-15] and [Katz 2009, pp. 633-636]. In Proposition III, located on page 5, Murphy took a “rational and integer function” of $$x$$, $$\varphi(x)$$, and supposed $$h$$ to be an indeterminate quantity. He showed that it was possible to find a series expansion for $$\varphi(x+h)$$:

$\varphi(x) + \varphi'(x) \cdot h + \varphi''(x) \cdot \frac{h^2}{1 \cdot 2} + \varphi'''(x) \cdot \frac{h^3}{1 \cdot 2 \cdot 3} + \&\,\,{\rm{etc.}}$

Then, using a direct application of the binomial theorem and rearrangement of terms, Murphy was able to find expressions for $$\varphi^{\prime}(x),\varphi^{\prime\prime}(x), \dots,$$ which he called the derived functions of $$\phi(x)$$ [1839, pp. 5-6]. The term “derived functions” came from Lagrange’s fonctions dérivées. Interestingly, this is where our modern term “derivative” comes from.

Within this work, Murphy incorporated increasingly popular ideas of the nineteenth century, such as limits and convergence. In his 1831 “On the Resolution of Algebraical Equations,” Murphy stated [1831, p. 125]:

The researches of Lagrange on the part of Pure Analysis, which forms the subject of the present Memoir, have been followed up with considerable success by many foreign Mathematicians, amongst whom M. Augustin Cauchy deserves to be particularly distinguished.

Therefore, it is clear that Murphy combined the work of Lagrange and Augustin-Louis Cauchy (1789-1857) to create his theory of pure analysis. Throughout Murphy’s book, readers will see the enlightened ideas of Cauchy [1821]. For example [Murphy 1839, pp. 7-8]:

This proposition shows that $$\varphi(x)$$, which is here used for a rational integer function of $$x,$$ is perfectly continuous, that is, while the results are always real, the difference $$\varphi(\alpha+h)−\varphi(\alpha)$$ and $$\varphi(\alpha-h)−\varphi(\alpha)$$ converge to zero, as $$h$$ is made to diminish continually towards the same, whatever $$\alpha$$ may be.

It is apparent that Murphy adopted Cauchy’s concept of a continuous function from his Cours d’analyse [Cauchy 1821, p. 26]:

The function $$f(x)$$ is a continuous function of x between the assigned limits if, for each value of $$x$$ between these limits, the numerical value of the difference $$f(x+\alpha)−f(x)$$ decreases indefinitely with the numerical value of $$\alpha$$.

After laying a foundation of pure analysis, Murphy then developed his theory of algebraic equations.

##### Overview of Algebraic Equations

Murphy presented a plethora of familiar rules and problems that fall under the scope of algebraic equations, most notably:

• Every equation of odd dimensions has necessarily a real root of a contrary sign to that which affects its last term.
• Every equation of even dimensions has necessarily two real roots, one positive, the other negative, provided the last term is negative.
• Every equation has either a real root or an imaginary couple.
• Descartes’ Rule: There cannot be more positive roots to an equation than there are alternations of signs in its successive terms, nor more negative than there are sequences of like signs.
• To find all the roots of the equation $${x^p}= 1$$ when $$p$$ is prime.
• Combined application of Sturm's Theorem, and Lagrange’s method of continued fractions, to numerical equations.

All of these rules and tasks occurred among the 106 "Article" (or section) headings in Murphy’s Treatise on the Theory of Algebraical Equations [1839, pp. v-vii, ix, xi].

Although his book contained results from other mathematicians, such as Charles-François Sturm (1803-1855), Fourier, and Lagrange, Murphy presented original material. For instance, he corrected one of Fourier’s propositions involving recurring series. (A recurring series is a sequence such that any given term is generated by the sum of a given number of previous terms each multiplied by given constants; that is, by a linear recurrence relation.) Recurring series were invented by Abraham De Moivre (1667-1754) and used by Daniel Bernoulli (1700-1782) in the approximation of solutions to algebraic equations [Euler 1770, pp. 419-420].) Let $$A,B,C,D,E,F,...$$ be a recurring series such that the quotients $$\frac{B}{A},$$ $$\frac{C}{B},$$ $$\frac{D}{C}, \ldots$$ converge to the greatest real root of a particular equation. According to Fourier [1830], the second series $AD−BC,\,BE−CD,\,CF−DE, . . .,$ which is derived from the first, converges to the sum of the roots of the proposed equation. Murphy stated that this theorem was incorrect, and proved that the quotients converge to the product of the real roots.

It is clear that Murphy achieved his objective: to provide a resource on the theory of algebraic equations. This was his final noteworthy contribution to mathematics before his death.

# Robert Murphy: Mathematician and Physicist - Murphy’s Works: Integral Equations

Author(s):
Anthony J. Del Latto (Columbia University) and Salvatore J. Petrilli, Jr. (Adelphi University)

### Murphy’s Works: Integral Equations

##### Overview of Integral Equations

The following works fall within Murphy’s “Integral Equations” research area:

• “On the General Properties of Definite Integrals” [1830],
• “First Memoir on the Inverse Method of Definite Integrals, with Physical Applications” [1832],
• Elementary Principles of the Theories of Electricity, Heat, and Molecular Actions. Part I on Electricity [1833],
• “On the Mathematical Laws of Electrical Influence” [1833],
• “Second Memoir on the Inverse Method of Definite Integrals, with Physical  Applications” [1833],
• “Third Memoir on the Inverse Method of Definite Integrals, with Physical Applications” [1835],
• “On a New Theorem in Analysis” [1837],
• “On the Composition of Two Rectangular Forces Acting on a Point”  [1837],
• “On Atmospheric Refraction” [1842].

Much of Murphy’s work on integral equations was centered around integral transformations. Using Murphy’s notation, a function $$\varphi(x)$$ that is equal to an integral of a function $$f(t)$$ with respect to some kernel function, initially $${t^x},$$ would be denoted as:$\phi(x)={\int^1_0}{f(t)}\,{t^x}\,dt$  [Cross 1985, p. 123]. (More generally, the kernel function, $$k(u,t),$$ is a function of variables $$u$$ and $$t$$ that allows us to perform an integral transform from a function $$f(t)$$ to a function $$g(u)$$.) Murphy’s goal was to determine the function $$f.$$ He adopted the limits $$0$$ and $$1$$ from Gauss’ Methodus nova integralium valores per approximationem inveniendi [1815].  Murphy referred to this problem as “an inverse method of Definite Integrals, by which we may re-ascend from the known integral, to the unknown function under the sign of definite integration” [1832b, p. 353].

Furthermore, Murphy considered both continuous and discontinuous functions in his theory of integral equations. According to Murphy, it was necessary to examine discontinuous functions because “the phænomena presented by nature are mostly of that kind” [Murphy 1832b, p. 355]. Essentially, Murphy’s examination of discontinuous functions was motivated by their applications in areas of physics, such as electrostatics, gravity, and heat [Murphy 1833a].

 Figure 5. Portrait of Pierre-Simon Laplace (Source: Public domain)
##### Contributions to Physics and Potential Research Projects

In addition to his work in mathematics, Murphy contributed to several areas of physics, such as the  subject of electricity. In his book,  Elementary Principles of the Theo ries of Electricity, Heat, and Molecular Actions. Part I on Electric ity, Murphy presented the theory of electricity. According to Cross [1985], Murphy’s book was intended to be a text for students studying  at Cambridge. At the start of his book, Murphy acknowledged that Volume III of  the Mécanique Céleste [1803] of Pierre-Simon Laplace (1749-1827) was “indispensable in  investigations respecting electricity”  [Murphy 1833a, p. v]. Consequently,  he introduced Laplace’s work in a section titled “Preliminary Propositions.” In general, Murphy discussed  Laplace functions, and this was the  first appearance of this term in English [Miller 2010]. Murphy expanded  on Laplace’s work and arrived at a  more general class of functions which  are of importance in the study of  the theory of latent electricity. According to Murphy himself, he “obtained several new and remarkable theorems with respect to Laplace’s functions” [Murphy 1833a, p. vi]. Interestingly, Murphy did not highlight what was “new” in his work because “that will easily be recognized by those who are already acquainted with the subject” [Murphy 1833a, p. vi]. An in-depth analysis of this work, to include revealing results original to Murphy, could serve as an interesting research project or potential thesis topic.

It is clear that Murphy made significant contributions to the fields of mathematics and physics, as demonstrated by his work on integral equations. Readers interested in a more in-depth analysis of Murphy’s work on integral equations and contributions to physics can refer to Cross [1985], Grattan-Guinness [1985], and Wilson [1985].

##### Reference to Servois

We end this section by noting a key citation that the authors previously were unaware existed. Murphy mentioned “functions of operation of the distributive kind (that is, such whose action on the whole, is the sum of the actions on the parts)” [1832b, p. 354] and referenced the work of François-Joseph Servois (1767-1847) on the calculus [Servois 1814]. Previously, there was no solid evidence that Murphy read Servois’ contributions to calculus, because Murphy made no direct reference to Servois in his “First Memoir on the Theory of Analytic Operations” [1837b], a paper in which Murphy used and expanded on many of Servois’ ideas. In the next section we provide an exposition of Murphy’s “First Memoir.”

# Robert Murphy: Mathematician and Physicist - Murphy’s Works: Operator Calculus

Author(s):
Anthony J. Del Latto (Columbia University) and Salvatore J. Petrilli, Jr. (Adelphi University)

### Murphy’s Works: Operator Calculus

##### Overview of Murphy’s Analytic Operators

In 1837, Murphy published a paper called “First Memoir on the Theory of Analytic Operations.” The purpose of Murphy’s paper was to present the theory of operators within calculus. Although the paper was titled the “First Memoir,” there were no follow-up papers.

Murphy began his paper by presenting key terminology and notation. For example, he introduced the term subjects to refer to objects on which operations are performed. Also, Murphy used the notation $$[f(x)]\psi$$ when discussing the operator $$\psi$$ applied to the function $$f(x),$$ where the subject is contained within brackets.

He then discussed special types of operators. He considered the operators $$p$$ and $$q$$ as fixed or free, where “in the first case a change in the order in which they are to be performed would affect the result, in the second case it would not do so” [Murphy 1837b, p. 181]. Furthermore, Murphy introduced the idea of a linear operator. According to Murphy [1837], “Linear operations in analysis are those of which the action on any subject is made up by the several actions on the parts, connected by the sign $$+$$ or $$−,$$ of which the subject is composed” [p. 181]. For example, let $$a$$ and $$b$$ be subjects and $$p$$ be the operation of multiplying by the quantity $$p.$$ Then $\left[a \pm b\right]p = \left[a\right]p \pm \left[b\right]p,$ which is a linear operator according to Murphy’s definition. Interestingly, Murphy was the first mathematician to use the term linear when referring to a special class of operators [Allaire and Bradley 2002].

 Figure 6. Portrait of George Peacock (Source: Public domain)
##### Contributions to Linear Operator Theory

Murphy expanded on previous work in the theory of linear operators, such as the research conducted by Servois. In his elaboration on the theory of linear operators, Murphy introduced the term appendage and defined the appendage of a linear operator to be “the result of its action on zero” [Murphy 1837b, p. 188]. Here, “action” means the inverse image of the operator. The appendage is analogous to the modern day kernel of a linear transformation, and this was the first time that the kernel of an operator had been examined [Allaire and Bradley 2002].

Murphy further expanded on his concept of the appendage. For example, he considered $$[P]\theta=0,$$ where $$\theta$$ is a linear operator and $$P$$ the subject connected with the nature of the operator. He then stated that $$P$$ would express a form. Here, the form will be the set of all elements which are mapped to $$0$$ by $$\theta.$$ It is reasonable to conjecture that Murphy adopted this term from the 1830 work, A Treatise on Algebra, by George Peacock (1791-1858).

Finally, Murphy considered the equation $$\theta \iota = \iota x,$$ where $$\theta,$$ $$\iota,$$ and $$x$$ are operators given without a subject and a general identity. He then defined $$\iota$$ to be intermediate with respect to $$\theta$$ and $$x$$. If you are given $$\theta$$ or $$x$$ and the intermediate $$\iota$$, then you can solve for the third operation; i.e., $$\theta = \iota x \iota^{-1}$$ and $$x = \iota ^{-1} \theta \iota$$. In modern algebra we would call $$x$$ the conjugate of $$\theta$$.  Murphy [1837b] demonstrated that intermediate operators are also intermediate between any operations that are the same functions of the extremes (here, the extremes are $$\theta$$ and $$x$$). Murphy accomplished this by demonstrating that if $$\theta i = i x$$, then $$\theta ^{n} i = i x^{n}$$. Essentially, Murphy examined the closure property of intermediate operations.

Throughout his paper, Murphy derived several results involving differentials, such as a method for expanding a function into a series. In his derivation, he created expansions that are equivalent to the “theorem of Taylor” [1837b, p. 183]. In addition, he discovered some “remarkable properties” of the operation $$\psi$$ which changes $$x$$ into $$x + h$$. Louis François Antoine Arbogast (1759-1803) called this operator the varied state [1800] and Murphy stated that this operator was free and linear.

Murphy then incorporated his concept of a “free” linear operator (the modern definition of a linear operator) into the calculus - an important contribution to the general theory of operators. Despite its significance, this paper was Murphy’s first and only work in the category of operator calculus.

# Robert Murphy: Mathematician and Physicist - Conclusion - Appendix

Author(s):
Anthony J. Del Latto (Columbia University) and Salvatore J. Petrilli, Jr. (Adelphi University)

### Conclusion

Murphy’s years of alcohol abuse took a toll on his health. In 1843, he contracted tuberculosis of the lungs [Barry 1999] and he died soon after, on March 12, 1843. It was shortly after Murphy’s death that De Morgan made the claim about his genius with which we opened this biography: “He had a true genius for mathematical invention” [Venn 2009]. Murphy was buried in Kensal Green Cemetery, London, where “[t]he grave has no headstone nor landing stone nor surround. It is totally unmarked” [Barry 1999, p. 173].

We end this biographical journey with Murphy’s obituary, which appeared in The Gentleman’s Magazine:

March 12. The Rev. Robert Murphy, M.A. Fellow of Gonville and Caius college, Cambridge, and Examiner in Mathematics and Natural Philosophy at University College, London. He took his degree of B.A. in 1829; and was the author of “Elementary Principles of the Theories of Electricity, Heat, and Molecular Actions” [Urban 1843, p. 545].

Figure 7. Likeness of Robert Murphy (ca. 1829) (Source: Permission granted by the Master and Fellows of Gonville and Caius College, Cambridge)

### Appendix

As noted above, all but the first of Murphy's known works are readily available from Google Books or JSTOR. This first work,

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 [1824],

was itself published in pamphlet form and was noticed by at least one well-known mathematician, Augustus De Morgan, in 1864 or earlier. The authors have provided a transcription of Murphy's Refutation, with commentary, as an appendix available here.

### Acknowledgment

The authors would like to thank Dr. Patricia R. Allaire for providing us with foundational material on Robert Murphy. We are also grateful to the Gonville and Caius College libraries, and particularly to Ms. Kate McQuillian, for locating many of the sources and pictures for us.

The authors are extremely grateful to an anonymous referee for his/her many helpful suggestions and corrections. Finally, we are thankful to Dr. Patricia R. Allaire who read the revision of this paper and made additional helpful suggestions.

Anthony J. Del Latto has a B.S. in mathematics from Adelphi University, where he served as a tutor for the Department of Mathematics and Computer Science from 2009 to 2012 and teacher’s assistant for the course MTH 457: Abstract Algebra during his senior year. He is currently pursuing an M.A. in mathematics education with initial certification for grades 7-12 from Teachers College, Columbia University.

Salvatore J. Petrilli, Jr. is an assistant professor at Adelphi University. He has a B.S. in mathematics from Adelphi University and an M.A. in mathematics from Hofstra University. He received an Ed.D. in mathematics education from Teachers College, Columbia University, where his advisor was J. Philip Smith. His research interests include history of mathematics and mathematics education.

# Robert Murphy: Mathematician and Physicist - End of a Legacy

Author(s):
Anthony J. Del Latto (Columbia University) and Salvatore J. Petrilli, Jr. (Adelphi University)

### The End of a Legacy

Murphy’s years of alcohol abuse took a toll on his health. In 1843, he contracted tuberculosis of the lungs [Barry 1999] and he died soon after, on March 12, 1843. It was shortly after Murphy’s death that De Morgan made the claim about his genius with which we opened this biography: “He had a true genius for mathematical invention” [Venn 2009]. Murphy was buried in Kensal Green Cemetery, London, where “[t]he grave has no headstone nor landing stone nor surround. It is totally unmarked” [Barry 1999, p. 173].

We end this biographical journey with Murphy’s obituary, which appeared in The Gentleman’s Magazine:

March 12. The Rev. Robert Murphy, M.A. Fellow of Gonville and Caius college, Cambridge, and Examiner in Mathematics and Natural Philosophy at University College, London. He took his degree of B.A. in 1829; and was the author of “Elementary Principles of the Theories of Electricity, Heat, and Molecular Actions” [Urban 1843, p. 545].

Figure 7. Likeness of Robert Murphy (ca. 1829). (Source: Permission granted by the Master and Fellows of Gonville and Caius College, Cambridge)

# Robert Murphy: Mathematician and Physicist - Bibliography

Author(s):
Anthony J. Del Latto (Columbia University) and Salvatore J. Petrilli, Jr. (Adelphi University)

### References

[Allaire 2002] Allaire, P. (2002). “Where was Robert Murphy 1833-1835? Or Did Murphy Meet George Green?,” Proceedings of Canadian Society for History and Philosophy of Mathematics, 15, 9 - 12.

[Allaire and Bradley 2002] Allaire, P. and Bradley, R. (2002). “Symbolic Algebra as a Foundation for Calculus: D. F. Gregory’s Contribution,” Historia Mathematica, 29, 395-426.

[Arbogast 1800] Arbogast, L. F. A. (1800). Du calcul des dérivations. Strasbourg: LeVrault Fréres.

[Barry 1986] Barry, D. (1986). “Robert Murphy: Mathematician of True Genius,” Mallow Field Club Journal, Issue 4, 5 - 11.

[Barry 1999] Barry, N. (1999). “Mallow’s Prodigy – Robert Murphy,” Mallow Field Club Journal, Issue 16, 157 - 175.

[Bonnycastle 1813] Bonnycastle, J. (1813). A Treatise on Algebra, in Practice and Theory, in Two Volumes, with Notes and Illustrations; containing a Variety of Particulars Relating to the Discoveries and Improvements that have been made in this Branch of Analysis (Third Edition). London: J. Johnson and Co. (Available on Google Books.)

[Bradley and Petrilli 2010] Bradley, R. E. and Petrilli, S. J. (2010). “Servois’ 1814 Essay on the Principles of the Differential Calculus, with an English Translation,” Convergence, 7:
http://www.maa.org/publications/periodicals/convergence/servois-1814-essay-on-the-principles-of-the-differential-calculus-with-an-english-translation

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[Gonville and Caius College Matriculation Book] Gonville and Caius College Matriculation Book. Reference number: TUT/01/01/02. Housed within the Archives of Gonville and Caius College, p. 546.

[Katz 2009] Katz, V. (2009). A History of Mathematics: An Introduction (Third Edition). Boston, MA: Addison-Wesley.

[Lagrange 1797] Lagrange, J. (1797). Théorie des fonctions analytiques. Paris: L’Imprimerie de la République. Second edition, Paris: Vve. Courciet, 1813.

[Laplace 1802] Laplace, P. S. (1802). Mécanique Céleste (Volume III). Paris: Gauthier-Villars.

[Long 1846] Long, G. (1846). “Murphy, Robert,” The Supplement to the Penny Cyclopædia of the Society for the Diffusion of Useful Knowledge, II, 337- 338.

[Miller 2010] Miller, J. (2010). Earliest Known Uses of Some of the Words of Mathematics. Retrieved July 14, 2011, from http://jeff560.tripod.com/mathword.html.

[Murphy 1824] Murphy, R. (1824). 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. Mallow: John Haynes, Printer, Spa-Walk. The authors have provided their transcription of this work, with commentary, as an appendix available here.

[Murphy 1830] Murphy, R. (1830). “On the General Properties of Definite Integrals,” Transactions of the Cambridge Philosophical Society, 3, 429-443. (Available on Google Books.)

[Murphy 1831] Murphy, R. (1831). “On the Resolution of Algebraical Equations,” Transactions of the Cambridge Philosophical Society, 4, 126-153. (Available on Google Books.)

[Murphy 1832a] Murphy, R. (1832). “On Elimination between an Indefinite Number of Unknown Quantities,” Transactions of the Cambridge Philosophical Society, 5, 65-75. (Available on Google Books.)

[Murphy 1832b] Murphy, R. (1832). “First Memoir on the Inverse Method of Definite Integrals, with Physical Applications,” Transactions of the Cambridge Philosophical Society, 4, 353-408. (Available on Google Books.)

[Murphy 1833a] Murphy, R. (1833). Elementary Principles of the Theories of Electricity, Heat, and Molecular Actions. Part I on Electricity. Cambridge: Pitt Press. (Available on Google Books.)

[Murphy 1833b] Murphy, R. (1833). “On the Existence of Real or Imaginary Root to Any Equation,” Philosophical Magazine, 2, 60-61. (Available on Google Books.)

[Murphy 1833c] Murphy, R. (1833). “On the Real Functions of Imaginary Quantities,” Philosophical Magazine, 2, 287-288. (Available on Google Books.)

[Murphy 1833d] Murphy, R. (1833). “On the Mathematical Laws of Electrical Influence,” Philosophical Magazine, 2, 350-351. (Available on Google Books.)

[Murphy 1833e] Murphy, R. (1833). “Further Development of the Existence of a Real or Imaginary Root to Any Equation,” Philosophical Magazine, 2, 220-221. (Available on Google Books.)

[Murphy 1833f] Murphy, R. (1833). “Second Memoir on the Inverse Method of Definite Integrals, with Physical Applications,” Transactions of the Cambridge Philosophical Society, 5, 113-148. (Available on Google Books.)

[Murphy 1835b] Murphy, R. (1835). “Third Memoir on the Inverse Method of Definite Integrals, with Physical Applications,” Transactions of the Cambridge Philosophical Society, 5, 315-393. (Available on Google Books.)

[Murphy 1835a] Murphy, R. (1835). “On the Resolution of Equations in Finite Differences,” Transactions of the Cambridge Philosophical Society, 6, 92- 106. (Available on Google Books.)

[Murphy 1837a] Murphy, R. (1837). “Analysis of the Roots of Equations,” Philosophical Transactions of the Royal Society of London, 127, 161-178. (Available on JSTOR.)

[Murphy 1837b] Murphy, R. (1837). “First Memoir on the Theory of Analytic Operations,” Philosophical Transactions of the Royal Society of London, 127, 179-210. (Available on JSTOR.)

[Murphy 1837c] Murphy, R. (1837). “On a New Theorem in Analysis,” Philosophical Magazine, 10, 28-32. (Available on Google Books.)

[Murphy 1837d] Murphy, R. (1837). “On the Composition of Two Rectangular Forces Acting on a Point,” Philosophical Magazine, 10, 105-108. (Available on Google Books.)

[Murphy 1837e] Murphy, R. (1837). “Remark on an Error of Fourier in his ‘Analyse des équations,” Philosophical Magazine, 10, 38-40. (Available on Google Books.)

[Murphy 1839] Murphy, R. (1839). A Treatise on the Theory of Algebraical Equations. London: The Society for the Diffusion of Useful Knowledge. (Available on Google Books.)

[Murphy 1841a] Murphy, R. (1841). “Remark on Primitive Radices’,” Philosophical Magazine, 19, 369. (Available on Google Books.)

[Murphy 1841b] Murphy, R. (1841). “Calculations of Logarithms by Means of Algebraic Fractions,” Philosophical Magazine, 18, 479-480. (Available on Google Books.)

[Murphy 1842] Murphy, R. (1842). “On Atmospheric Refraction,” Philosophical Magazine, 20, 310-312. (Available on Google Books.)

[O’Connor and Robertson 1999] O’Connor, J. and Robertson, E. (1999). “Doubling the Cube.” Retrieved March 16, 2012, from MacTutor History of Mathematics Archive,
Web site: http://www-history.mcs.st-and.ac.uk/HistTopics/Doubling_the_cube.html.

[Peacock 1830] Peacock, G. (1830). A Treatise on Algebra. London: Cambridge.

[Petrova 1978] Petrova, S. S. (1978). “The Origin of the Linear Operator Theory in the Works of Servois and Murphy,” History and Methodology of the Natural Sciences, 20, 122-128. (Unpublished translation by Valery Krupkin.)

[Robertson 1848] Robertson, J. C. (Ed.) (1848). “Robert Murphy, The Mathematician,” The Mechanics’ Magazine, London: Robertson and Co., XLIX. 354-356.

[Servois 1814] Servois, F. J. (1814). “Essai sur un nouveau mode d’exposition des principes du calcul différentiel,” Annales de mathématiques pures et appliqués, 5 (1814-1815), 93-140.

[Urban 1843] Urban, S. (1843). “Clergy Deceased,” The Gentleman’s Magazine, XIX, 545.

[Venn 2009] Venn, J. (2009) Biographical History of Gonville and Caius College, 1349-1897, 2, General Books LLC.

[Whiston 1791] Whiston, W. (1791). The Elements of Euclid: with Select Theorems out of Archimedes. By the Learned Andres Tacquet. To which are added, Practical Corollaries, shewing the Uses of many of the Propositions. Dublin: R. Jackson in Meath-Street. (Available on Google Books.)

[Wilson 1985] Wilson, D.B. (1985). “The Educational Matrix: Physics Education at Early-Victorian Cambridge, Edinburgh and Glasgow Universities.” In P. M. Harman (Ed.), Wranglers and Physicists: Studies on Cambridge Mathematical Physics in the Nineteenth Century (pp. 12-48). Manchester: Manchester University Press.

[Wood 1830] Wood, J. (1830). The Elements of Algebra: Designed for the Use of Students in the University (Ninth Edition). Cambridge: J. Smith, Printer to the University. (Available on Google Books.)