According to Piccolino [1984], the nineteenth century is considered by many historians to be a Golden Age in the development of mathematics. Advancements in several branches of mathematics, such as geometry and analysis, occurred during this revolutionary time period. Another area that experienced change was algebra [Piccolino 1984]. Prior to the nineteenth century, algebra usually referred to the theory of solving equations; however, by 1900 it involved the study of mathematical structures, such as groups, rings, and fields [Katz 2009]. Mathematicians found that these structures often did not share properties found in the real and complex number systems, such as commutativity. This fundamental change in algebraic thought is characteristic of modern or abstract algebra.

**Figure 1.** A stamp issued in the Soviet Union in 1983 portrayed Al-Khowârizmî (public domain).

One of the earliest instances where we see a mathematician's work on the solvability of algebraic equations is in the writings of Mohammed ibn Mûsâ al-Khowârizmî (ca. 790 CE - ca. 850 CE). Al- Khowârizmî provided solutions to linear (first-degree) equations and quadratic (second-degree) equations, but his results were presented verbally, without the use of algebraic symbols [Dunham 1990]. Al-Khowârizmî did not recognize either negative coefficients or negative solutions in his general solution to the quadratic equation \(ax^2 + bx + c = 0\), which he broke up into six cases. According to David Eugene Smith [1958], the first significant treatment of negative numbers was by Girolamo Cardano (1501-1576) in his 1545 book on algebra, *Ars Magna.* The first consideration of imaginary solutions occurred a few years later when Rafael Bombelli (1526-1572) used imaginary numbers as a “tool” for solving cubic equations [Dunham 1990, pp. 150-151]. According to Victor Katz, Bombelli's work “provided mathematicians with the first hint that there was some sense to dealing with” imaginary numbers in their algebraic work [2009, p. 407].

**Figure 2.** Joseph-Louis Lagrange (public domain).

Initial developments in abstract algebra occurred in Continental Europe during the late eighteenth and early nineteenth centuries. These changes were driven by problems in classical algebra, such as the solvability of third, fourth, and higher degree equations [Piccolino 1984]. The works of Joseph-Louis Lagrange (1736-1813), Augustin-Louis Cauchy (1789-1857), Paolo Ruffini (1765-1822), Niels Henrik Abel (1802-1829), and Evariste Galois (1811-1832) were of central importance during this time period and contained several concepts associated with modern group theory.

**Figure 3.** Augustin-Louis Cauchy. Painting by Jean Roller ca. 1840 (public domain).

Lagrange explored the solvability of equations via the theory of permutations. Cauchy also made contributions to the theory of permutations by introducing concepts such as the identity permutation, a permutation that does not change a given arrangement of objects. Furthermore, Ruffini made several attempts at proving that the general equation of degree five is unsolvable in terms of radicals. Although Ruffini's efforts were not successful, his work provided a foundation for Abel's proof that such a solution cannot exist [Katz 2009]. Finally, Galois made significant contributions to the theory of solvability of algebraic equations by studying the structure of algebraic equations, particularly what he called “the group of the equation” [Katz 2009, p. 726]. We also owe to Galois the first known use of the term “group” in mathematics, which appeared in 1830 [Boyer 1989].

Although the work on algebraic solvability was carried out on the Continent, it was the British school of algebra that was primarily responsible for the shift in algebraic thinking towards abstract structural properties. As we shall see, this was not necessarily done by building on continental work on algebraic solvability, but rather by extending properties of ordinary arithmetic and of what we would call functions from analysis. Important figures in this movement included George Peacock (1791-1858), Duncan Farquharson Gregory (1813-1844), and William Rowan Hamilton (1805-1865). Peacock introduced the notions of arithmetical algebra and symbolical algebra. He defined arithmetical algebra as a universal arithmetic (using letters instead of numbers) of positive numbers [Katz 2009]. In this system, the term \(a - b\) had meaning only if \(a\) was greater than or equal to \(b\). On the other hand, symbolic algebra referred to the study of operations that were defined through arbitrary laws. In Peacock's symbolic algebra, \(a - b\) was valid regardless of the relationship between the symbols \(a\) and \(b\) [Piccolino 1984]. However, his laws in symbolic algebra were derived using principles found in his arithmetical algebra [Katz 2009]. Peacock was on the cusp of formulating an internally consistent algebra and his efforts in that direction were extended by Gregory.

Gregory, founder of the *Cambridge Mathematical Journal,* focused on algebraic structure. In his works, he often referred to the ideas of commutativity, distributivity, index operations (a sort of law of exponents for operators), and inverses, which he described as “circulating operations.” He also mentioned the principle of the separation of symbols of operation, crediting the French mathematician François-Joseph Servois (1767-1847) as the first to “correctly give” the procedure [Allaire and Bradley 2002, p. 410]. Gregory appears to have been one of the first mathematicians to establish a connection between differentiation in calculus and the ordinary symbols of algebra, noting that the commutative and distributive laws hold true for what he referred to as the symbols of differentiation. Despite his contributions to the development of abstract algebra, Gregory, like Peacock, maintained the stance that results in symbolic algebra had to suggest results in arithmetical algebra [Piccolino 1984].

**Figure 4.** Duncan Farquharson Gregory (public domain).

A new, internally consistent algebraic system was finally introduced by Hamilton with his discovery of quaternions on October 16, 1843. Hamilton extended the algebra of number pairs to ordered quadruples of numbers, \((a, b, c, d)\), and defined the quaternions as ordered quadruples of numbers that followed several rules [Katz 2009]. Most notably, Hamilton's quaternions did not satisfy the commutative postulate for multiplication [Boyer 1989]. His system was the first algebra that did not follow all of the laws established by Peacock [Katz 2009]. The freedom and structure present in Hamilton's system was unprecedented and, as a result, many historians consider his discovery of the quaternions as the beginning of abstract algebra [Piccolino 1984].

Fundamental structures in abstract algebra, such as groups and fields, were formally defined in the later part of the nineteenth century. Heinrich Weber (1842-1913) was the first mathematician to present detailed, axiomatic definitions of groups and fields [Katz 2009]. Weber's definition of a finite group was slightly different than the one that most mathematicians are familiar with today. His definition included three axioms analogous to the modern day ideas of closure, associativity, and left- and right-hand cancellation laws. The terminology of closure under an operation is first found in Saul Epsteen and J. H. Maclagan-Wedderburn's “On the Structure of Hypercomplex Number Systems,” which appeared in *Transactions of the American Mathematical Society,* Vol. 6, No. 2. in April of 1905 (see [Epsteen and Maclagan-Wedderburn 1905] and [Miller 2010]). Weber then showed that his three laws imply the existence of a unique identity element, and for each element, the existence of a unique inverse. He also incorporated his notion of group in the definition of a field. He defined a field as a set with two operators, addition and multiplication. In Weber's field, the entire set forms a commutative group under addition and the nonzero elements form a commutative group under multiplication as well. Weber also noted several properties of fields including the distributive law, which states that \(a\cdot (b + c) = a\cdot b + a\cdot c\) for all elements in the field [Katz 2009].

It is clear that mathematicians of the nineteenth century were concerned with foundational issues that spanned across several different areas of mathematics. Servois was no different and concentrated mainly on the foundational issues of the calculus. His attempts to settle the foundational issues of calculus were not successful [Bradley and Petrilli 2010]; however, as we will see, his work had a direct influence on the development of abstract algebra, and in particular, linear operator theory.