### 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.