American Scientist

Arthur Cayley: Mathematician Laureate of the Victorian Age. Tony Crilly. xxiv + 609 pp. Johns Hopkins University Press, 2006. $69.96.

James Joseph Sylvester: Jewish Mathematician in a Victorian World. xvi + 461 pp. Johns Hopkins University Press, 2006. $69.96.

Historian George Sarton often said that science advances in darkness, invisible to the majority of people, who are more interested in battles and other noisier activities. In his 1957 book The Study of the History of Mathematics, Sarton went on to say that if the history of science is secret, then the history of mathematics is doubly so, "for the growth of mathematics is unknown not only to the general public, but even to scientific workers."

Sarton's words help us understand why few have ever heard of Arthur Cayley (1821-95) or James Joseph Sylvester (1814-97), two of the most profound and prolific mathematicians of the Victorian era. Cayley's seminal investigations of matrix algebra, which constituted only a tiny portion of his 967 papers, were crucial for the development of linear algebra. The terms matrix, determinant and Jacobian, familiar to most science students, were invented by Sylvester, an enthusiastic poet who called himself the "mathematical Adam."

It is not clear when Cayley and Sylvester first met, but by 1847 they were corresponding to share thoughts about mathematics. They rallied and inspired each other for nearly four decades, drawn together by their common circumstances. Each had triumphed on the University of Cambridge's fearsome Tripos examinations: Sylvester placed second in 1837 and Cayley first in 1842. These accomplishments qualified them for mathematics professorships, but such positions were few and openings rare. Cayley was a Trinity College fellow at Cambridge for a few years and could have remained so, teaching routine courses and living a communal life. But had he married, he would have been compelled to resign his fellowship and then perhaps scratch out a clergyman's living at one of the parishes belonging to Trinity, far from the academic community. Sylvester's options were fewer. As a Jew, he could not even collect his Cambridge degree, since in order to do so he would have had to subscribe to the Thirty-Nine Articles of the Church of England.

 Cayley and Sylvester would travel other avenues. By 1846 both were studying law at London's Inns of Court, and each stayed on there for several years, Cayley drafting legal documents as a barrister-at-law and Sylvester working as an actuary and barrister. The first had escaped the stone, parental grip of the Church of England. The second was its unloved bastard.

Turning away legal and actuarial work, hoarding every spare minute for mathematics, Cayley and Sylvester communicated with each other about their research almost daily, building a new theory of algebraic invariants. So intertwined were their lives as mathematicians, both during and after their careers at the Inns of Court, that when E. T. Bell wrote his famous (and flawed) history, Men of Mathematics, he christened them the "Invariant Twins." It is appropriate that Johns Hopkins University Press should publish their biographies simultaneously.

Arthur Cayley: Mathematician Laureate of the Victorian Age, by well-known historian of mathematics Tony Crilly, is the first full-length account of Cayley's life. The book draws on many primary sources. Crilly begins every chapter with a brief abstract and divides the material into short sections, with subheads that help keep the reader oriented. He includes surprisingly few mathematical formulas and integrates his explanations of these smoothly with the biographical material.

Cayley was born into a comfortable family. His father, Henry, was a "Russia merchant," and Arthur spent the first seven years of his life in St. Petersburg, where his family enjoyed gracious living and high culture among British expatriates, at the price of harsh winters and isolation. The family returned to England in 1828.

Young Cayley attended King's College London, where his abilities were quickly recognized, and then entered Trinity College, Cambridge, in 1838—the year of Queen Victoria's coronation.

When a biographical subject has led a calm, pleasant life, it is the biographer who must suffer, searching for engaging matter. Fortunately for Crilly, the Victorian society in which Cayley moved was close-knit and lively. George Boole, Charles Dickens, Francis Galton, James Clerk Maxwell and William Thomson (later Lord Kelvin) are but a few Cayley intimates found in these pages. (Crilly includes such details as that Cayley disliked Dickens's novels, preferring instead those of Jane Austen.)

Anyone interested in the emerging role of the research mathematician in England will find Crilly's book particularly rewarding. Cayley finally attained a professorship at the age of 41, when he became the first person elected to the Sadlerian chair of mathematics at Cambridge. In accepting the position, he happily traded income for leisure.

Scientific research had by that time gained acceptance at Cambridge. However, pure mathematics was regarded as mere mental gymnastics, or at best a scientific tool; "mixed mathematics" (what today we call applied mathematics) was valued more. Shortly after Cayley was appointed, William Thomson expressed that popular bias to Hermann von Helmholtz: "Oh! that the CAYLEYS would devote that skill that they have to such things [Kirchoff's work on electrical conducting plates] instead of to pieces of algebra which possibly interest four people in the world, certainly not more, and possibly also only the one person who works."

Cayley was elected president of the British Association for the Advancement of Science in 1883, an honor that no pure mathematician had enjoyed in nearly 40 years. The association's annual meeting in Southport attracted scientists, politicians, preachers and the working class. Fear of mathematics being nothing new, an anxious rumor spread there that the new president would require a blackboard and a supply of chalk! Cayley's portrait was painted, and his address was reported in the Times.

A look back at the meeting's proceedings confirms that there were very few presentations in pure mathematics. The reader is left with the sad impression that despite England's pride in its "mathematician laureate," Cayley might have been better appreciated in France or Germany.

Cayley's lifelong friend Sylvester was loquacious and quick-tempered, Cayley's emotional antipode. Sylvester has been a focus of research for historian and mathematician Karen Parshall for more than two decades. In her 1998 book James Joseph Sylvester: Life and Work in Letters, she examined his mathematics with skill and insight. Sylvester sprinkled all of his writing with witty reflections, philosophical digressions and even lines of verse. Readers of his letters have developed a thirst to know more about him, and with the arrival of Parshall's masterful biography, James Joseph Sylvester: Jewish Mathematician in a Victorian World, they can imbibe at last.

Parshall turns over the political and religious soils that alternately starved and nourished Sylvester's work. Mathematics fertilizes her account, but she distributes it sparingly. What grows is a rich story of a stubborn genius willing to fight the world so that he might bestow his gifts.

We are introduced to Sylvester as a troubled child who is becoming increasingly aware of the anti-Semitism that will impede him. We later watch his hopes soar and then plummet as he struggled with the reality that in the 19th century few academic positions, particularly those at Oxford and Cambridge, could be achieved by an English Jew. After college, he had an unhappy stint for a few years as a professor of natural philosophy at secular University College London. In his mid-20s, he resigned the position to take another, across the Atlantic at the University of Virginia.

By offering an invitation to one of England's rising scholars, the University of Virginia hoped to elevate its own prestige. By accepting, Sylvester hoped to fashion there the sort of research professorship that he had longed to find at home. The two daydreams combined into a nightmare. When he arrived in 1841, the students, who were sons of the Southern aristocracy, cheered him. They welcomed the great man with an "illumination" in his honor, burning candles between stolen barrels of tar and, according to a contemporary account, "raising an unholy racket." When the students' fantasy of an Oxbridge don was spoiled by "a little, bluff, beef-fed English cockney, perfectly insignificant in his appearance, and raw and awkward in his manners," their admiration soon gave way to insubordination and threats of violence. That Sylvester acquired a "sword cane" for protection, and that he was later attacked by a pair of brothers, are very likely, Parshall concludes. That Sylvester drew it and struck a nonfatal blow, as a student reported, remains an unconfirmed legend.

Back in England, Sylvester anticipated a bleak future. He had no job and had accomplished little mathematically in the preceding two years. In December 1844 he accepted an offer of a position as an actuary at the newly created Equity and Law Life Assurance Society at Lincoln's Inn Fields in London. To do so must have been painful for someone who wanted so much to do research as a mathematician. A couple of years later he began studying for the bar.

Sylvester's business efforts and his intense mathematical collaboration with Cayley during this phase of his life were successful but exhausting. When a professorship at the Royal Military Academy in Woolrich became available in 1855, one that did not require taking an oath "on the true faith of a Christian," he applied, and with the help of an influential friend, Lord Brougham, he got the position. The Academy was more concerned with training future officers than with producing pure mathematics. Nonetheless, for 15 years Sylvester persisted in struggling to mold his position to suit his aspirations, until in 1869 he was forced into mandatory retirement.

Seeking solace, he turned to poetry, publishing Laws of Verse (1870), an unreadable study of versification that prompted one reviewer to advise the author to concentrate instead on mathematics. Over the next six years, Sylvester kept busy in London with a variety of activities that included music and public speaking. It seemed impossible for him to attain another academic position.

In 1876, salvation came from America in the form of an offer from the new Johns Hopkins University to lead a graduate program in mathematics and launch the American Journal of Mathematics, a research journal that continues today. For the next seven years, Sylvester would be surrounded by enthusiastic students and appreciative colleagues. While he was there, the political and social scenery back in England was being transformed as new rights were won by working men, the emerging middle class and Jews.

At age 70, Sylvester was elected to the Savilian professorship of geometry at Oxford. He wiped away tears as he left behind his grateful students and colleagues in Baltimore. It is difficult for us to appreciate fully the emotions that he felt as he was finally embraced by Oxford. "She is a good dear mother our University here," he wrote to Cayley, "and stretches out her arms with impartial fondness to take all her children to her bosom even those she has not reared at her breast."

Cayley and Sylvester differed wildly in personality and background. But mathematically they were often of a single mind. Together they were pioneers, breaking free of the powerful attraction exerted by Newton, who was at the center of the solar system that was English mathematics. The abstract algebra that Cayley and Sylvester developed, so useless in the opinion of Thomson, would prove essential to Werner Heisenberg, Albert Einstein and other future scientists. Both of these biographies will be appreciated by anyone who is amused by this cosmic irony.