Life and Works of Arthur Cayley

William Li

Math 10, Fall 2011

Mr. Pavitch

December 12, 2011

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Life and Works of Arthur Cayley

Arthur Cayley was born to Henry Cayley and Maria Antonia on August 16th, 1821 at Richmond in

Surrey (Crilly 3). Arthur spent his early childhood living in Russia and would occasionally spend summers

back in England. His idyllic childhood was spent in great comfort and joy, though often in absence of

many friends. His father, part of the 930-man-strong Russia Company, was a merchant who made a living

in St. Petersburg. This group of semi-wealthy expatriates was the link to the growing trade between both

countries (Crilly 6-7). When Cayley was 10, the family moved to Blackheath, a district in London, so he

could begin his formal education under George Potticary’s preparatory school at Eliot Place. From an

early age, Cayley demonstrated an affinity for mathematics, especially a liking towards arithmetical

calculations. Though not indicative of prodigal genius, his love of numbers is certainly pertinent to his

character as a whole. One of his teachers at Eliot Place remarked that Cayley would often ask “for sums

in Long Division to do while the other little boys were at play” (Crilly 16). In August 1835, at the age of

fourteen, Arthur enrolled in King’s College in London, a private school for the growing wealthy middle-

class, where his great skill in mathematics became apparent. He was advised to pursue mathematics

rather than continue his father’s line of work (“Cayley Biography”). This became a pivotal moment in

Cayley’s life. In May 1838, prior to receiving the prestigious Silver Medal for Chemistry, often reserved for

scientists and researchers, he was admitted into Trinity College. Henry Cayley was disappointed that his

son chose an academic career, rather than following in his footsteps, as did the long line of Cayleys

before him. The entire Cayley clan was a class of merchants and businessmen, and, having grown up in

this environment, where fathers passed down their occupations to their son, Henry Cayley found it difficult

to accept his son’s decision. However, he was eventually persuaded to relent, and Arthur Cayley went on

to enter Cambridge University (Crilly 26). Cayley would spend the next four years under the tutorage of

George Peacock, John Moore Heath, and William Hopkins (Crilly 32-33). Cayley soon rose to the top,

quickly, achieving the excellence and prestige that many of his contemporaries, several years his senior,

could not. Arthur Cayley graduated as Senior Wrangler, winning a Fellowship, and proceeded to teach.

However, the position did not pay well, and needing a means to support his continued studies and

research in mathematics, he decided to become a lawyer. Over the course of the next fourteen years,

Cayley thrived in the legal profession, but, true to his roots, published over two hundred fifty mathematical

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papers. In 1863, a new position opened up at Cambridge: the Sadleirian professor, whose position

description stated that the individual would “explain and teach the principles of pure mathematics and to

apply himself to the advancement of that science” (“Cayley Biography”). Cayley was appointed. Cayley,

after so many years of law and conveyancing, was happy to fully devote his time to the subject he loved

more than anything else, eagerly accepting the fact that his position would pay many times less than what

he earned as a lawyer (Bell 383).

One of Cayley’s greatest contributions to the study of linear algebra is perhaps the idea of

matrices. Although matrices were not his creation, matrix algebra is the subject most commonly

associated with his name. Though James Joseph Sylvester, Cayley’s close friend, was the one to coin the

term “matrix”, Arthur Cayley’s “A Memoir on the Theory of Matrices” is what popularized the notation and

explained how to do basic arithmetic with matrices. One of the most important theorems he created for

this subject is now called the Cayley-Hamilton theorem, which he cooperated with William Hamilton to

create. The theorem states that an n x n square matrix A satisfies its own characteristic polynomial, which

is defined as

The theorem proposes that , for example, if

then

Applying the Cayley-Hamilton theorem, this would mean that , which is true (Crilly

227-228). Cayley is also known for discovering Cayley’s transform, an “orthogonal transformation that

solves a variant of the Cayley-Hermite problem… of use in quantum mechanics” (Crilly 471). In the field of

geometry, Cayley is known for his work with n-dimensional geometry, where he was able to unify metrical

and projective geometry. His work on matrices is invaluable to the foundation for quantum mechanics,

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developed by Werner Heisenberg in 1925. (Bell, 399) Even more important is his joint development of the

theory of algebraic invariance, which analyzes functions and groups which are not altered by a

transformation under some matrix group, with James Sylvester while they both worked in law (Bell 390-

392).

Today, Cayley’s work has spanned into various different fields. For example, in 1998, Sarah

Flannery devised the Cayley-Purser algorithm, a public-key cryptography algorithm which boasted to be

many times faster than the standard RSA encryption. It was based off the idea that matrix multiplication is

non-commutative, something which Cayley outlined, though the algorithm was eventually found to be

insecure.

Arthur Cayley is without a doubt one of the most impressive pure mathematicians of the

nineteenth century. He began from modest and humble beginnings and eventually became well respected

in his circle of friends, all extremely successful and accomplished already. His invaluable contributions

extend beyond pure mathematics, going into fields such as astronomy and mechanics as well. He is also

the most prolific writer of his time, releasing over nine hundred papers in thirteen quarto volumes. His

devotion and love for mathematics allowed him to write three hundred papers alone in the fourteen years

he worked as a lawyer. Even as his health failed, he continued to review his old works and inspired

himself to write down even more new ideas, through the days leading up to his death. Cayley never had a

chance to fulfill his dream of founding a research school of mathematics, but his work has nonetheless

inspired countless others and his legacy lives on in the influential theorems and ideas he left behind.

Bibliography

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Bell, E. T. Men of Mathematics. Harmondsworth: Penguin, 1937. Print.

“Cayley Biography.” MacTutor History of Mathematics. Web. 11 Dec. 2011. <http://www.history.mcs.st-

andrews.ac.uk/Biographies/Cayley.html>

Crilly, A. J. Arthur Cayley: Mathematician Laureate of the Victorian Age. Baltimore, MD: Johns Hopkins

UP, 2006. Print.