# Arthur Cayley

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 a great many areas of mathematics from analysis , algebra , geometry to astronomy and mechanics, but is best known for his role in the introduction of the abstract group concept .

## Life

Cayley was the son of the merchant Henry Cayley, whose ancestors came from Yorkshire , but who settled in Saint Petersburg , where Cayley lived as a child for eight years. In 1829 the family moved back to England, to Blackheath near London, where he received private lessons. Cayley attended King's College in London from the age of 14, where his teacher recommended studying mathematics at Cambridge because of his talent. He studied from 1838 at Trinity College , Cambridge , where he excelled in Greek, French, German, Italian and mathematics. In mathematics was his Philosopher George Peacock , and Cayley was in the Tripos exams 1842 Senior Wrangler , as students already published three papers in the Cambridge Mathematical Journal (whose themes from his study of the works of Joseph-Louis Lagrange and Pierre-Simon Laplace revealed ) and won the Smith Prize. In 1845 he obtained his master’s degree. He also won a fellowship from Trinity College in a competitive exam, stayed in Cambridge for four years and published several papers during this time, but then had to look for a more lucrative job. He decided to become a lawyer and joined Lincoln's Inn in London in 1846. While still a lawyer, he traveled to Dublin to hear William Rowan Hamilton lectures on quaternions . Cayley worked mostly as a notary. With his friend James Joseph Sylvester , who worked as an insurance broker, he continued to discuss mathematics and published around 250 mathematical essays in his 14 years as a lawyer.

In 1863 he was appointed to the newly established Sadlerian Chair of Pure Mathematics in Cambridge. That was a significant loss of income for Cayley, but it meant the fulfillment of his lifelong dream. Simultaneously with the acceptance of the professorship he married in 1863. In 1872 he became an honorary fellow of Trinity College and in 1875 a fellow. In 1882 he lectured in Baltimore at the invitation of New Year's Eve at Johns Hopkins University . In 1883 he became President of the British Association. From 1889 his collected works appeared at Cambridge University Press, which at the end comprised 13 quart volumes and 967 works. He edited the first seven volumes himself, the following volumes his successor as Sadlerian Professor Andrew Russell Forsyth .

The Cayley Purser algorithm , the asteroid (16755) Cayley and the Cayley crater on the moon are named after Arthur Cayley .

In 1852 he was elected as a member (" Fellow ") in the Royal Society , which in 1859 awarded him the Royal Medal and in 1882 the Copley Medal . In 1863 he became a corresponding member of the Académie des sciences . On December 4, 1865 he was elected an Honorary Fellow of the Royal Society of Edinburgh . In 1866 he was elected to the American Academy of Arts and Sciences , in 1883 to the National Academy of Sciences . Cayley also received the De Morgan Medal from the London Mathematical Society and the Huygens Medal in Leiden. He has received multiple 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, Saint Petersburg, Milan , Rome , Leiden, Uppsala and Budapest. He was an officer in the French Legion of Honor. He was intermittently President of the Cambridge Philosophical Society, the London Mathematical Society and the Royal Astronomical Society. In 1874 his portrait, painted by Lowes Dickinson, was hung in the hall of Trinity College and his bust was also hung in the library of Trinity College during his lifetime.

Cayley was also a passionate mountaineer.

## plant

Cayley founded the invariant theory with Sylvester , an area that both dominated so much in England that they were also called the "invariant twins". Cayley introduced the term (and name) of the abstract group in 1854 , to which he not only assigned the permutation groups, which have otherwise been much investigated since Augustin Louis Cauchy , but also, for example, matrices and quaternions . He used multiplication tables to define the groups. Cayley's precursors in defining the group concept were Cauchy and Evariste Galois , who, however, only dealt with permutation groups. Galois did not explicitly define groups either, but Cayley was familiar with his work (which had been re-published by Liouville in 1845). Cayley also wrote on matrices, determinants , quaternions, and algebraic equations . He found Cayley's theorem, which is important in algebra . Independently of John Thomas Graves , he discovered octonions (a divisional algebra ), also called Cayley numbers , in 1845 .

In the dispute over the use of Hamilton's quaternions, which was waged in England at the end of the 19th century, he defended the use of coordinates against Hamilton's avid supporter Peter Guthrie Tait in 1894 : the quaternion is a beautiful concept, but its applications are less so.

Cayley also produced a projective model of non-Euclidean ( hyperbolic ) geometry (Cayley-Klein model), in which the straight lines are straight line segments inside a circular disk with a distance (metric) that exceeds the double ratio of two (used in projective geometry ) Points with the end points of the straight line section laid through them is formed on the edge of the circle.

His work on algebraic geometry, for example on the singularities of algebraic curves and the classification of cubic curves, was also important. Like Sylvester, Cayley was a pioneer of graph theory (terms like Cayleygraph and Cayleybaum were named after him). The formula from graph theory for the number of trees with labeled nodes comes from him . This is called Cayley's formula and says: if there are n nodes, these are${\ displaystyle n ^ {(n-2)}}$

He only published one book during his lifetime.

## Works by Arthur Cayley with available digital copies

• An elementary treatise on elliptic functions . Bell, Cambridge / Deighton 1876
• The collected mathematical papers of Arthur Cayley . Volume 1. University Press, Cambridge 1889-1897; archive.org
• The collected mathematical papers of Arthur Cayley . Volume 2. University Press, Cambridge 1889-1897; archive.org
• The collected mathematical papers of Arthur Cayley . Volume 3. University Press, Cambridge 1889-1897; archive.org
• The collected mathematical papers of Arthur Cayley . Volume 4. University Press, Cambridge 1889-1897; archive.org
• The collected mathematical papers of Arthur Cayley . Volume 5. University Press, Cambridge 1889-1897; archive.org
• The collected mathematical papers of Arthur Cayley . Volume 6. University Press, Cambridge 1889-1897; archive.org
• The collected mathematical papers of Arthur Cayley . Volume 7. University Press, Cambridge 1889-1897; archive.org
• The collected mathematical papers of Arthur Cayley . Volume 8. University Press, Cambridge 1889-1897; archive.org
• The collected mathematical papers of Arthur Cayley . Volume 9. University Press, Cambridge 1889-1897; archive.org
• The collected mathematical papers of Arthur Cayley . Volume 10. University Press, Cambridge 1889-1897; archive.org
• The collected mathematical papers of Arthur Cayley . Volume 11. University Press, Cambridge 1889-1897; archive.org
• The collected mathematical papers of Arthur Cayley . Volume 12. University Press, Cambridge 1889-1897; archive.org
• The collected mathematical papers of Arthur Cayley . Volume 13. University Press, Cambridge 1889-1897; archive.org

## literature

• Andrew Russell Forsyth (Ed.): The Collected Mathematical Papers of Arthur Cayley . 13 volumes. Cambridge University Press, 1889-1897
• Tony Crilly: A Victorian mathematician: Arthur Cayley (1821-1895) . In: The Mathematical Gazette , Volume 79, 1995, pp. 259-262.
• Tony Crilly: Arthur Cayley: Mathematician Laureate of the Victorian Age . Johns Hopkins University Press, 2006
• Crilly: Arthur Cayley: The Road not Taken . In: Mathematical Intelligencer , Volume 20, 1998, pp. 49-53
• Jeremy Gray : Arthur Cayley (1821-1895) . In: The Mathematical Intelligencer , Volume 17, Issue 4, 1995, p. 62
• Max Noether : Arthur Cayley . In: Mathematische Annalen , Volume 46, 1895, pp. 462-480.
• Cayley, Arthur . In: Encyclopædia Britannica . 11th edition. tape 5 : Calhoun - Chatelaine . London 1910, p. 589 (English, full text [ Wikisource ]).
• Alexander MacFarlane: Lectures on ten British mathematicians of the 19th century . With chapter on Cayley; archive.org

## Individual evidence

1. ^ List of members since 1666: Letter C. Académie des sciences, accessed on October 28, 2019 (French).
2. ^ Fellows Directory. Biographical Index: Former RSE Fellows 1783–2002. (PDF) Royal Society of Edinburgh, accessed October 16, 2019 .
3. For example in his series of articles Memoirs upon Quantics (with “quantics” meant algebraic forms) in 10 parts from 1854 to 1878
4. So the chapter heading on Cayley and Sylvester in the well-known collection of biographies by Eric Temple Bell Men of Mathematics
5. Cayley: On the theory of groups, as depending on the symbolic equation${\ displaystyle \ Theta ^ {n} = 1}$ . In: Philosophical Magazine , 1854, Volume 7, pp. 40-47, reprinted in Collected Works , Volume 2, pp. 123-130
6. He was one of the first to deal with it: On the theory of linear transformations . In: Cambridge Mathematical Journal , Volume 4, 1845, pp. 193-209
7. For the story, see the article on the group concept at MacTutor . As Hans Wussing demonstrated, he also referred to Galois by the name group.
8. Literally: I have the highest admiration for the notion of a quaternion; but, as I consider the full moon far more beautiful than any moonlit view, so I regard the notion of a quaternion as far more beautiful than any of its applications , Proceedings of the Royal Society of Edinburgh, 1894
9. Cayley: A Sixth Memoir upon Quantics . In: Philosophical Transactions of the Royal Society of London, Volume 159, 1859, pp. 61-91
10. See the distance function in the article Hyperbolic Geometry
11. Cayley: A theorem on trees . In: Quarterly Journal of Mathematics , Volume 23, 1889, pp. 376-378. Several proofs can be found in Aigner, Ziegler: The BOOK of Evidence . Springer publishing house
12. An Elementary Treatise on elliptic functions . 1876; 2nd edition: G. Bell, London 1895