# Arthur Cayley

Arthur Cayley ( born August 16, 1821 in Richmond upon Thames, Surrey, † January 26, 1895 in Cambridge ) was an English mathematician. He dealt with many areas of mathematics of calculus, algebra, geometry to astronomy and mechanics, but is primarily known for his role in the introduction of the abstract group concept.

## Life

Cayley was the son of businessman Henry Cayley, whose ancestors came from Yorkshire, but settled in St. Petersburg, where Cayley lived for eight years as a child. 1829 the family moved back to England, to Blackheath in London, where he was taught privately. Cayley visited over 14 years, the King's College in London, where his teacher recommended a study of mathematics at Cambridge because of his talent. He studied from 1838 at Trinity College, Cambridge, where he distinguished himself in Greek, French, German, Italian and mathematics. In mathematics was his tutor George Peacock, and Cayley was in the Tripos examinations 1842 Senior Wrangler, even as freshmen published three papers in the Cambridge Mathematical Journal ( whose themes arose from his study of the works of Joseph -Louis Lagrange, Pierre- Simon Laplace ) and Smith won the prize. In 1845 he received his master's degree. He also won in a competitive examination was elected a Fellow of Trinity College, remained four years in Cambridge and has published several works during this time, but then had to find a more lucrative profession. He decided to become a lawyer and in 1846 joined Lincoln's Inn, London. During his legal education, he traveled to Dublin to hear lectures by William Rowan Hamilton on Quaternions. Cayley worked mainly as a notary. With his friend James Joseph Sylvester, who worked as an insurance broker, but he discussed further on mathematics and published in his 14 years as a lawyer around 250 mathematical papers.

In 1863 he was appointed to the newly established Sadlerian - Chair of Pure Mathematics at Cambridge. This was a significant loss of income for Cayley, but meant the fulfillment of his ambitions. Simultaneously with the acceptance of the professorship he married 1863. 1872 he was Honorary Fellow of Trinity College and 1875 Fellow. In 1882 he lectured in Baltimore at the invitation of Sylvester at Johns Hopkins University. In 1883 he became president of the British Association. From 1889 his collected works by Cambridge University Press, which included 13 quart volumes and 967 work at the end appeared. The first seven volumes he was still out for yourself, the following volumes of his successor as Sadlerian Professor Andrew Russell Forsyth.

After Arthur Cayley Cayley - Purser of the algorithm and the Cayley craters are named on the moon.

In 1852 he was elected as a member ( "Fellow" ) to the Royal Society, in 1859, the Royal Medal in 1882 and the Copley Medal awarded him. Cayley also received the De Morgan Medal of the London Mathematical Society and the Huygens Medal in Leiden. He was much honorary doctorates (including Oxford, Dublin, Göttingen, Heidelberg, Leiden, Bologna, Edinburgh). Cayley was a corresponding member of the Institut de France, the academies in Berlin, Göttingen, St Petersburg, Milan, Rome, Leiden, Uppsala and Budapest. He was an officer of the French Legion of Honour. He was temporarily President of the Cambridge Philosophical Society, the London Mathematical Society and the Royal Astronomical Society. In 1874 his portrait, painted by Lowes Dickinson, hung in the hall of Trinity College and his bust also alive in the library of Trinity College.

Cayley was also an avid mountaineer.

## Work

Cayley -founded with Sylvester invariant theory, an area that both so much dominated in England that they were called the " invariants Twins ". Cayley introduced in 1854 the concept ( and name ) of the abstract group, which he not only about assigning else since Augustin Louis Cauchy much studied permutation groups, but also, for example, matrices and quaternions. For the definition of the groups he used multiplication tables. Precursor of Cayley in the definition of the concept group were Cauchy and Evariste Galois, however, only treated permutation groups. But Galois groups defined not explicitly, Cayley knew his work ( which was re-released in 1845 by Liouville ). Cayley also wrote of matrices, determinants, quaternions and algebraic equations. He has found the key in the Cayley algebra. Regardless of John Thomas Graves, he was called in 1845 discovered the octonions ( a division algebra ), and Cayley numbers.

In the dispute over the use of Hamilton's quaternions, which was conducted at the end of the 19th century in England, he defended in 1894 against Hamilton's zealous partisans of Peter Guthrie Tait to use the coordinates: namely the quaternion is a nice concept, but their applications less.

From Cayley comes also a projective model of non-Euclidean ( hyperbolic ) geometry ( Cayley-Klein model), in which the straight line segments in the interior of a circular disk are at a distance ( metric ), the two on the ( used in projective geometry ) double ratio points is formed with the end points of the line segment specified in them on the edge of the circle.

Of significance were his work on algebraic geometry, for example of the singularities of algebraic curves, and the classification of the cubic curves. How Sylvester was also a pioneer of the Cayley graph theory ( concepts like Cayleygraph and Cayleybaum were there named after him). From him the formula is derived from graph theory for the number of trees with labeled nodes. This is called Cayley 's formula and states that if n nodes are the

During his lifetime he published only one book ( An Elementary Treatise on Elliptic functions 1876, 2nd edition, G. Bell, London 1895).