The Artistic, Algebraic, and Severe “Russian” Genius Arthur Cayley

Arthur Cayley, unlike most of his contemporaries, contributed to almost every nineteenth-century mathematical subdiscipline. His wide-ranging and productive career presented a difficulty in the outpouring of obituaries written by friends and colleagues in 1895. How to classify Cayley as a mathematician? The conundrum extended beyond determining whether Cayley was more of an algebraist or a geometer to increasingly labyrinthine divisions. By focusing on these divisions and comparisons in a series of brief biographical memorials, classifying markers of his relative positions emerge.¹

Metabiographical approaches to the history of mathematics serve as a window onto “normative statements about values and virtues” [27, p. 81].² The most notorious example of biographical discord for the history of mathematics may be the case of Évariste Galois, whose brief life left much space for speculation. In turn, Galois’s biographical legacy has served as a tool for better understanding the legend of mathematical genius as well as the study of algebraic equations in the nineteenth century.³ Cayley’s biography is much less dramatic, but it still affords opportunities for subjective assessment. In addition to the obituaries written in 1895, I will also draw on classifications of Cayley in the several biographical sketches preceding his death.

An early attempt to capture the essence of Cayley appeared on the occasion of his election as president of the British Association for the Advancement of Science. His friend and collaborator George Salmon provided “some slight sketch” as a biography for the “Science Worthies” column in Nature [25, p. 481].

In assessing Cayley’s contributions, Salmon declared that his “subject is the life of a great artist who has had courage to despise the allurements of avarice or ambition, and has found more happiness from a life devoted to the contemplation of beauty and truth than if he had striven to make himself richer, or otherwise push himself on in the world” [25, p. 481].⁴ This affectionate description served to situate Cayley firmly as a pure mathematician.

Even more specifically, though Salmon and Cayley had worked together on geometric topics, Salmon accorded with “Mr. Glaisher, who has described [Cayley] as the greatest living master of algebra” [25, p. 483]. Salmon acknowledged that “no part of mathematics comes amiss to him,” but concluded that “he is always happiest when he can translate his theorems into pure algebra and show that a proposed result is but the expression of an algebraical fact.” In this aspect, Salmon contrasted Cayley with the recently deceased “arithmetical” H. J. Smith.

Salmon further divided mathematicians (and chess players) “into the book-learned and the original” [25, p. 483]. Cayley was both, combining “great knowledge of books with the power to strike into new paths of their own.” Alongside his praise, Salmon admitted that Cayley could be “difficult to read,” because his papers followed a synthetic method, that is, he “usually begins by trying to establish at once the highest generalisation he has reached, writing down equations and proceeding to make calculations as to the good of which he has not taken his readers into his confidence.”

So, Salmon presented Cayley to his audience of British scientists as an artist (not interested in wealth or popularity) and an algebraist (not arithmetical). He also described him as well read and original, but, in following a synthetic (not analytic) method in his writing, generally hard to read.

A very different image emerged from Gaston Darboux’s review of the first volume of Cayley’s collected mathematical papers for the Bulletin des Sciences Mathématiques in 1893 [7]. Darboux praised Cayley as a nineteenth-century Leonhard Euler, who “has studied and made progress in all parts of mathematics, from number theory in its highest branches to astronomy in its most difficult parts” [7, p. 141].⁵ While most mathematicians could be divided between geometers and analysts, “M. Cayley is neither a pure geometer nor a pure analyst, he is at once one and the other.”⁶ Moreover, reading Cayley is “easy, where nothing is left in the shadows, not even the subjects of new works nor the deepest points” [7, p. 142].⁷

That same year, in Felix Klein’s Evanston Colloquium lectures, there appeared another set of “main categories,” among mathematicians as “logicians, formalists, and intuitionists” [15, p. 2]. Formalists, who “excel mainly in the skillful formal treatment of a given question, in devising for it an ‘algorithm,’ ” were exemplified by Paul Gordan, Cayley, and James Joseph Sylvester. As a border case, Alfred Clebsch was both a formalist and an intuitionist—“those who lay particular stress on geometrical intuition (Anschauung), not in pure geometry only, but in all branches of mathematics.”

The challenge of sorting Cayley in his lifetime resonated with those tasked with summing him up after death. Writing again on Cayley for the Cambridge Review in 1895, J. W. L. Glaisher declared that there was “scarcely any department in the whole range of mathematical literature which was outside the field of Cayley’s work,” with the only exception “the Philosophy of Mathematics, and the similar class of subjects associated in this country with De Morgan’s name” [13, p. 174].⁸ Like Darboux, Glaisher observed a likeness between Cayley and Euler. The two had “the same power, fertility and exuberance, and both have left their mark on every branch of science that existed at the time.”

However, Glaisher noted “a great difference between the graceful luxuriance and pleased enthusiasm of Euler’s style, and the cold and rugged severity of Cayley’s” [6, p. 174]. Glaisher attributed these aspects of Cayley’s writing in part to “his legal training,” in which there was awarded “very little importance to mere elegance of form or simplicity of development” [6, p. 175]. Still, he found Cayley’s results “beautiful” despite their appearance, noting that “many a reader who has at length mastered a difficult memoir of his must have longed to rewrite it in a form which would show out its merits to greater advantage.” Glaisher clearly preferred a carefully composed memoir in which the author had paused between investigation and publication. He posited that Cayley’s writing style could be a consequence of his “natural tendency of the mind” toward truth; results “existing in rerum naturæ” needed no adornment.

When Charlotte Angas Scott wrote for the Bulletin of the American Mathematical Society in February 1895, she noted that “very contradictory opinions [had] been expressed as to the effect of Cayley’s style” [26, p. 141].⁹ In an effort at resolution, Scott divided style into two aspects. Invoking Darboux, she commended the “cumulative force” of Cayley’s mathematical style, or “the marshalling of the subject-matter,” and explained how Cayley must be read—“there is no going rapidly over it, with the hope of getting a superficial idea, leaving difficulties to be considered at a more convenient season; it must be all or nothing.” The prepared reader would find “no real difficulties beyond the great preliminary one of the subject itself.” Scott also found much to appreciate in Cayley’s literary style—“simplicity and directness itself.” Cayley’s writing had neither Plücker’s “hasty intuitiveness” and “repulsive” style [26, p. 135] nor “the seductive literary appeal” of Salmon [26, p. 141].

Rather than claiming Euler as Cayley’s forerunner, however, Scott looked for analogous nineteenth-century figures. Citing the obituary of Alfred Clebsch, written by his friends and published in the Mathematische Annalen in 1873 [2], Scott introduced “two main lines” of mathematical thought—“the one concerning itself with the precise defining and justifying of the fundamental conceptions of the science” and “the other accepting a certain small number of fundamental conceptions and working from these” [26, p. 136]. For Scott, the first group was represented by Gauss, Dirichlet, and research in the theory of numbers and the theory of functions. She assigned Jacobi, Plücker, and Cayley to the “latter division” and pointed to Cayley’s lack of sympathy with Riemann’s ideas as indicative of his relative disinterest in foundational questions—adding further nuance to Glaisher’s exception of the philosophy of mathematics.

Scott’s invocation of Clebsch’s obituary as well as her identification of Julius Plücker as “an impulse starting Cayley’s genius along certain lines” cannot be found in other British or American descriptions of Cayley [26, p. 135]. However, Max Noether, among the several friends who had collectively composed Clebsch’s obituary in 1874, similarly drew out the connection between Cayley and Plücker in his piece on Cayley for the Mathematische Annalen in June 1895 [20].

By aligning Cayley with Plücker, Scott also challenged Klein’s analysis of mathematicians. Scott determined “a shade of reproach” in the “not altogether pleasing” second category of formalists, which Klein had seen as particularly prevalent in the United States [26, p. 140]. While Scott concurred that Cayley excelled in algorithmic solutions, she contested the adverb “mainly.” Indeed, some have been “perhaps dazzled by his supremacy over the rest of the world in analysis” and thus passed “lightly over the geometrical side of his investigations.” Though Klein’s categories cut through disciplinary divisions of mathematics, Scott’s counterargument to Cayley as formalist hinged on his attention to geometry. Scott mused “that while Cayley’s processes were algebraic, since the language of algebra was simpler to him than the ordinary language of words, the color of his thought was essentially geometrical.” However, she also conceded that “the passages which convey to one mind the idea of a geometrical substratum may very well convey to another the idea that nothing existed for Cayley but the analysis.”

Andrew Forsyth, writing for the Proceedings of the Royal Society in June 1895, made a similar argument. He agreed with Salmon that “Cayley was specially happy in the treatment of algebraical developments,” but then cautioned that “an inadequate estimate of his genius would be obtained by supposing that he was almost entirely an analyst” [12, pp. xxvi–xxvii]. Forsyth recalled how “much of his thinking, not a little of his writings, is completely geometrical; and his contributions to line geometry, his introduction of the Absolute into geometry, his continued recurrence to the methods in pure geometry invented by Poncelet and Chasles, should be sufficient to range him among geometricians.”

Forsyth likewise spoke to the “varying opinions” as to Cayley’s style [12, p. xxvi]. There was agreement that “it is not easy to skim one of his papers; any attempt to do so leads to an inadequate estimate of what it usually establishes.” But there were variations among obituarists in the “preconceived views of what a mathematical paper should be.” For Forsyth, it was “not difficult to read one of his papers, even to grasp the contents well, provided proper care be devoted to it, because difficulties that occur are completely solved, and nothing lies in the background to cause doubt or suggest incompleteness.” Moreover, “his literary style is direct, simple and clear” [12, p. xxvii]. However, these attributes could also be read as “severe and present a curious contrast to the luxuriant enthusiasm which pervades so many of Sylvester’s papers.” Even though Cayley never went “out of his way in order to secure beautiful form for the presentation of results,” some of “his papers are so admirably written that they satisfy the exacting critics.” Forsyth singled out Cayley’s “Sixth Memoir on Quantics” as one that “could not be presented in more attractive form.”

Cayley had published over nine hundred papers, and those responsible for summarizing his life and work depended on categories and analogies as tools of abbreviation. Despite these helpful intentions, putting mathematicians in boxes could end badly, as exemplified by the afterlife of Klein’s claim for the different natural abilities among mathematicians of the Teutonic, Latin, and Hebrew races [15, p. 46].¹⁰ If Klein’s division of mathematicians foreshadowed the application of racial types in mathematics in the twentieth century, the obituaries by George Bruce Halsted and Charles Sanders Peirce remind us that the 1890s were also susceptible to racist pseudoscience.¹¹

In April 1895, the American Mathematical Monthly published a biography of Arthur Cayley by Alexander MacFarlane [17] followed immediately by a biography by Halsted [14]. Both authors noted that Cayley’s mother was Russian. To Halsted, this fact was meaningful: “Cayley’s mother tongue was Russian, and his features had a Russian cast. Like so many Russians, he spoke most European languages well” [14, p. 103]. Then, throwing in some peculiar phrenological speculations, Halsted asked, “Can there be anything in what has so often been cited as fact, that in the Russian race alone the brain of the woman equals that of the man in size and weight?” Halsted’s aside is confusing, but suggestive. The equal relative weight of Russian women’s brains is posited as explanatory of Cayley’s mathematical success—something inherited from his mother’s side. At the end of his biography, he returned to the Russian heritage, citing Cayley’s “mother’s compatriot Lobachevsky” [14, p. 106].

That Cayley’s mother was not Russian did not prevent this detail from being a critical feature in some accounts of Cayley’s life story.¹² For the Evening Post, Peirce explained that “his physiognomy and cast of mind had much that was Russian” and explained what such an adjective denoted [9, pp. 644–645]:

His physiognomy was most striking. The shape of the head was extraordinary—rather flat, and seeming to have a cornice above it; the intensity of the gray eyes, which seemed to be looking through whatever opaque thing might be near to him, was unparalleled. The handsome but rather small nose and the perfectly unaffected and smilingly watchful mouth seemed boyish in their unsophisticated interest in things. But what a curious catlike intensity, as if just pouncing on a truth!

Figure 1.

Arthur Cayley over time. (Drawing by the author.)

Cayley’s outward appearance was a mirror of his thoughts (Figure 1). For Peirce’s Cayley, “the world of sense ... was miserably restricted and only half the truth,” while “geometrical figures, and even the substance of pure algebraical expressions, were far more truly real than anything in everyday life, whether of wood, of gold, or of flesh and blood” [9, p. 647]. With his unearthly head (inside and out), it is surprising that he was “twice married” and a blessing that he had “no children” [9, p. 646].

Cayley’s daughter, Mary, received a copy of Peirce’s obituary sent by her friend Mary Augusta Scott, an American academic who had studied English literature at Newnham College, University of Cambridge. Mary Cayley thanked Scott, noting “there are some mistakes in the facts, but the author writes warmly about my Father’s work” [5]. Meanwhile, as Thomas Fiske would later reminisce, Peirce had “infuriated Charlotte Angas Scott” with his unsupported statement “that Cayley had inherited his genius from a Russian whom his father had married in St. Petersburg” [11, p. 16]. The rumor persists. Cayley’s English-language Wikipedia page states that “according to some writers,” Maria Antonia Doughty, mother of Cayley, was Russian, “but her father’s name indicates an English origin.”

Obituaries of Cayley offer multidimensional spaces into which any mathematician might be plotted. Cayley—a mathematician with a long career, overwhelming productivity, and wide-ranging curiosity—was a particularly difficult case. This did not prevent his friends, colleagues, and students from trying to sort him even as they critiqued the inadequacy of previous attempts. The fact that much of Cayley’s research had been “overtly classificatory” may have been a further impetus [6, p. 193]. Despite the disciplinary expertise in classification among Victorian mathematicians, it should not be surprising that when they turned to study human beings, they were perfectly willing to work with imprecise definitions.

Salmon, so praised for his own prose, found fault with Cayley’s writing. Scott, dedicated exclusively to geometrical research, identified Cayley as a geometrician. Noether, a German, positioned Cayley in a German lineage in algebraic geometry. Halsted, a champion of non-Euclidean geometry, situated Cayley in that field in a Russian tradition.¹³

The debate around Cayley’s style centered on normative claims about mathematical practices.¹⁴ As Forsyth noted, these assessments were “views of what a mathematical paper should be” [12, p. xxvi]. Forsyth and Scott accepted that there was value in the necessary commitment required to read Cayley. Salmon and Glaisher longed for a more inviting form. Both groups identified writing mathematics as a literary activity, and both held that mathematical texts could be appreciated as literature. These comments demonstrate concern for a mathematical aesthetic, “a non-epistemic value that, however, is intrinsic to mathematics” [19, p. 253]. Following Reviel Netz’s aesthetic typology of Greek geometry texts, the lively debate around Cayley’s writings suggest that an analysis of his research publications could offer fruitful insight into standards and sources of mathematical beauty at the end of the nineteenth century. But as Cayley had warned in his presidential address to the British Association for the Advancement of Science, “as for anything else, so for a mathematical theory—beauty can be perceived, but not explained” [4, p. 449].