Algebraic Formalism within the Works of Servois and Its Influence on the Development of Linear Operator Theory - Servois' Influence on the Development of Linear Operator Theory

Servois' influence on the development of Linear Operator theory can be traced through the works of several well-known mathematicians, including Murphy and Gregory. Although the formalization of symbolic algebra is generally credited to these two mathematicians, we see several ideas associated with Linear Operator Theory in An Essay on Algebraic Development Containing the Principle Expansion in Common Algebra, in the Differential and Integral Calculus, and in the Calculus of Finite Differences; the General Term Being in Each Case Immediately Obtained by Means of a New and Comprehensive Notation by a lesser-known academic, Thomas Jarrett (1805-1882). Jarrett was an English cleric, Professor of Arabic at the University of Cambridge, and a linguist. According to the biography by E. J. Rapson [1892], he knew at least twenty different languages and would translate Chinese characters into Roman characters using a system that he devised himself. There is no biographical information that indicates that he had any formal mathematical training.

In the Preface to his book, Jarrett stated that part of his work was taken from the following mathematicians: Servois, Louis François Antoine Arbogast (1759-1803), John Frederick William Hershel (1792-1871), Carl Friedrich Hindenburg (1741-1808), Sylvestre François Lacroix (1765-1843), Pierre-Simon Laplace (1749-1827), Ferdinand Franz Schwiens (1780-1856), and Josef-Maria Hoëné-Wronski (1776-1853). Interestingly, Wronski's calculus was based on infinitesimals and Jarrett used no such foundations. Jarrett went on to say that some of the material was partly original; however, according to his biographers [Rapson 1892], the original contributions could simply have been new notation. Their contention is supported by Jarrett himself: “In the present Work [Algebraic Notation] is applied to the demonstration of the most important series in pure Analysis” [Jarrett 1831, p. III].

Jarrett's work is similar to Servois' “Essai” in that they both used algebra as a foundation for calculus; however, Jarrett presented Servois' material in a more structured format. Interestingly, when Jarrett discussed the summation operator he distinguished between operators (which he called operations) and functions, but when he presented his theory of the calculus he made no such distinction and classified both as functions. He derived many of the same results as Servois, only using different notation. Jarrett's calculus was based on the concept of the separation of symbols, which he credited to Servois in his Preface: “The demonstration of the legitimacy of the separation of the symbols of operation and quantity, with certain limitation, belongs to Servois ...” [Jarrett 1831, p. III]. At the heart of Jarrett's theory were Servois' distributive and commutative properties:

1. If \(\varphi (u)\) is such a function of \(u\) that \(\varphi (u + v) = \varphi (u) + \varphi (v)\), then \(\varphi (u)\) is called a distributive function of \(u\)
2. If \(\varphi (u)\) and \(\psi (u)\) are such functions of \(u\) that \(\varphi \psi (u) = \psi \varphi (u)\), then the functions \(\varphi (u)\) and \(\psi (u)\) are said to be commutative with each other.

From a modern standpoint, Jarrett defined a field in a fashion similar to Servois’ by introducing the notions of an identity, inverses, and closure, in addition to these two properties. Using properties of this field he derived his theory of the differential and integral calculus.

We also see Servois' ideas in the work of another mathematician, Robert Murphy (1806-1843). Murphy's [1837] “First Memoir on the Theory of Analytic Operations” is a detailed exposition on the theory of operators. Murphy clearly distinguished between functions and operations, and called the objects on which operations are performed subjects [Allaire and Bradley 2002]. With respect to his notation, if Murphy wanted to discuss the operator \(\psi\) applied to the function \(f(x)\), then he denoted it as \([f(x)] \psi\), where the subject is contained within brackets.

Murphy [1837] began his paper by examining 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 1837, p. 181]. To relate this to Servois' work, a free operator would be one that satisfied Servois' commutative property. Now, 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 makes \(p\) (or multiplication by \(p\)) a linear operator according to Murphy's definition. Thus, a linear operator is one that satisfies Servois' distributive property. In modern day mathematics, in order for \(p\) to be a linear operator it would also have to be free with respect to a constant \(k\). Additionally, Murphy was the first mathematician to use the term linear to describe a special class of operators [Allaire and Bradley 2002].

Bradley and Allaire [2002] state that Murphy derived many of the same results as Servois, only with greater clarity and brevity. However, Murphy also expanded on the theory of linear operators. For example, he defined the appendage of a linear operator as “the result of its action on zero” [Murphy 1837, p. 188]. Here, “action” refers to the inverse image of the operator. In modern day mathematics, the appendage would be referred to as the kernel of a linear transformation, and this was the first time that the kernel of an operator had been considered [Allaire and Bradley 2002]. The kernel is an important concept in understanding the behavior of linear transformations.

Unlike Jarrett, Murphy did not acknowledge a debt to Servois nor is there solid evidence that he actually read his work. Murphy's opening sections are devoted to the important properties of operators – that is, Servois' commutativity and distributivity – but Murphy gave these properties different names. However, Servois began the study of linear operators and his work was read in England, as demonstrated by Jarrett's use of his research in 1831.  As we will see, Duncan Gregory gave Servois the credit he was due.

According to Piccolino [1984], Gregory played a vital role in the development of symbolic algebra and was a key contributor to the overall advancement of mathematics in England during the late 1830s and early 1840s. In addition to his mathematical demonstrations, Gregory often provided philosophical insights to reinforce his views on algebra. For example, Gregory considered symbolic algebra as “the science which treats of the combinations of operations defined not by their nature, that is, by what they are or what they do, but by the laws of combination to which they are subject” [Gregory 1865, p. 2]. Essentially, Gregory believed that the general principles of algebra must fit a certain structure, which he called a class.

Throughout his Mathematical Writings [1865], Gregory provided several examples illustrating the “laws of combination” to which operations are subject. For example, he considered two classes of operations \(F\) and \(f\), which are connected by the following laws:

1. \(FF(a) = F(a)\)
2. \(ff(a) = F(a)\)
3. \(Ff(a) = f(a)\)
4. \(fF(a) = f(a)\)

His most general interpretation of these laws was multiplication for positive and negative numbers [Allaire and Bradley 2002]. For instance, (2.) shows that a negative number multiplied by a negative number yields a positive number. In this case, \(F\) and \(f\) should not be interpreted as functions, but rather as operations.

Next, he considered a general class of operations, which satisfy the following laws:

1. \(f(a) + f(b) = f(a + b)\)
2. \(f_1 f(a) = f f_1 (a)\)  

Gregory credited Servois with the classification of these laws, writing, “Servois, in a paper which does not seem to have received the attention it deserves, has called them, in respect of the first law of combination, distributive functions, and in respect to the second law, commutative functions” [Gregory 1865, pp. 6-7]. Whereas Murphy [1837] considered the special class of functions satisfying (1.) to be linear operators, Gregory noticed that these two laws together constitute a special class of operations, which are called linear transformations or linear operators in modern mathematics.

Gregory then provided an example to demonstrate his first law, where \(f\) is taken to be the operation of multiplying by a constant \(a\): \[a(x) + a(y) = a(x + y).\] In his Mathematical Writings, Gregory stated that Cauchy sometimes utilized the “laws of combination” [p. 7], so Gregory may have been familiar with the fact that this is the only continuous function that satisfies the distributive law.

Finally, Gregory defined a class of operations by the law \[f(x) + f(y) = f(xy).\] This is the first time we see a general law for a class in which two different operations are considered. Gregory related this definition to a familiar law of logarithms, that \(\ln (x) + \ln (y) = \ln (xy)\), saying “when \(x\) and \(y\) are numbers, the operation is identical with the arithmetical logarithm” [Gregory 1865, p. 11].  

Figure 7. Augustus De Morgan (public domain).

Petrova [1978] stated that linear operator theory began with Servois and was continued by Murphy. According to Allaire and Bradley [2002], Gregory was the next key figure in the development of this theory. We wondered, however, if any other mathematicians were influenced by the work of Servois. The authors examined numerous works written by mathematicians during the Golden Age of mathematics, including Peacock, Augustus De Morgan (1806-1871), and George Boole (1815-1864). These mathematicians made great advances in algebra; however, they appear to have made no significant contributions to the theory of linear operators. After examining several of their works, we now make some observations regarding the influence Servois may have had on these mathematicians. It should be noted that this analysis is highly subjective, because even though some of these mathematicians used methods similar to Servois', none gave him direct credit. Additionally, a majority of these works began to appear in the 1840s, and the works of Murphy and Gregory were already available by this time.

According to O'Connor and Robertson [1996], Peacock was interested in making reforms to Cambridge mathematics and he aided in the creation of the Analytical Society in 1815 as a result. The society was intended to bring the continental methods of the calculus to Cambridge. The reform began when Peacock translated Lacroix's calculus text, Traité élémentaire de calcul differéntiel et du calcul intégral [1802]. Using Lacroix's ideas, Peacock published his Collection of Examples of the Applications of the Differential and Integral Calculus [1820]. Lacroix's work was based on the calculus of Lagrange. Consequently, Peacock adopted many of the methods presented by Lacroix. Peacock did not explicitly use Servois' methods in his work and made no claim about the algebraic properties of operators. However, because Peacock was interested in the continental calculus, it is possible that he was familiar with the works of Servois.  

Now, De Morgan, who was a student of Peacock's, presented the algebraic definitions for distributivity and commutativity in his work, Trigonometry and Double Algebra. These definitions are very similar to the ones that students would learn in a high school algebra course today. For instance, De Morgan stated, “A symbol is said to be distributive over terms or factors when it is the same thing whether we combine that symbol with each of the terms or factors, or whether we make it apply to the compound term or factor” [De Morgan 1849, pp. 102-103]. Being a student of Peacock's, De Morgan could have been familiar with the works of the continental mathematicians. This is further supported by the fact that he used the terms distributive and commutative in a fashion similar to Servois'.

Figure 8.  George Boole (public domain).

Finally, in his Treatise on the Calculus of Finite Differences [1860], Boole gave the laws for the symbols \(\Delta\) and \(\frac{d}{dx}\). For instance, he stated:

1. The symbol \(\Delta\) is distributive in its operation. Thus, \(\Delta \left(u_x + v_x + \&c. \right) = \Delta u_x + \Delta v_x \cdots\).
2. The symbol \(\Delta\) is commutative with respect to any constant coefficients in the terms of the subject to which it is applied. Thus \(a\) being constant, \(\Delta a u_x = a \Delta u_x\).

Interestingly, Boole defined a special case of Servois' “commutative” property, where linear operators commute with constant factors.

Since Boole was a student of Gregory's, it is reasonable to conjecture that he was introduced to the works of Servois via Gregory's teachings. Servois' influence can be seen in Boole's own statements about \(\Delta\) and \(\frac{d}{dx}\) being distributive and commutative operators.