Сборник материалов международной научной конференции студентов, магистрантов, аспирантов

А.П. Кондратюк, В.В. Хильчук

Республика Беларусь, Брест, БрГУ имени А.С. Пушкина

Научный руководитель – П.Н. Резько

ARTHUR CAYLEY AND HIS ALGEBRA OF MATRICES

Arthur Cayley was a British mathematician who helped to found the British school of pure mathematics. He is most famous for developing the algebra of matrices and non-Euclidean and n-dimensional geometry. Cayley worked primarily with algebra and published several papers on mathematics. Even as a child, Cayley loved solving mathematical problems. He enjoyed the subject and was also an avid reader. He pursued his studies at Trinity College, Cambridge, and entered the field of mathematics. Cayley was the first person to figure out that Euclidean geometry was a special case of projective geometry. He also dipped into the field of astronomy and mechanics. He was motivated and inspired by Karl Jacobi and wrote a book called “An Elementary Treatise on Elliptic Functions” based on these studies. Cayley was also fluent in French, German, Greek and Italian. He hypothesized the Cayley–Hamilton theorem, and his legacy leaves behind many such works including the Cayley’s theorem, Cayley algebra and even Cayley’s Ù process.

Cayley was born in Richmond, England on 16th August 1821 to Henry Cayley and Maria Antonia Doughty. His brother Charles Bagot Cayley was a linguist. His father, a merchant, settled with his family in Saint Petersburg. For the first eight years, Cayley grew up in Germany.

In 1829 they moved back to England and settled in Blackheath, near London. Cayley was sent to a private school for four years, and even at his young age, he had an affinity for mathematics. At the age of fourteen, he was sent to King’s College School. His genius in mathematics was duly noted and his teacher told him to pursue mathematics instead of falling into his father’s footsteps. In 1838, at the tender age of seventeen, Cayley began learning at Trinity College, Cambridge. He finished his undergraduate course with honours, in 1842. He was awarded the “Senior Wrangler” in “Mathematical Tripo” and first place in the competition for the “Smith Prizes”. After this, he continued his studies by taking up an M.A. degree and won a fellowship. He resided in Cambridge after winning a fellowship. During this period, he gave lectures on mathematics. His first contribution was made in 1841, to the “Cambridge Mathematical Journal” established by Gregory and Robert Leslie Ellis. He submitted three papers on subjects based on reading the “Mécanique analytique de Lagrange” and some of the works of “Laplace”. His main works, however, were the twenty-eight memoirs to the “Mathematical Journal”.

At the end of seven years, Cayley ended his fellowship at Cambridge and had to choose a profession. Cayley opted for law and joined Lincoln’s Inn, London. During these fourteen years, Cayley wrote about 200–300 papers.

In 1863, he accepted the Sadleirian professorship in mathematics at Cambridge. He devoted himself to mathematical research and gave up on his career in law. When it came to mathematics, Cayley was known to be an individualist. His skill was formidable and it was said that he had an intuitive understanding of mathematical theories. Cayley made valuable contributions to algebraic theory of curves and surfaces, graph theory, group theory, linear algebra, combinatorics and elliptic functions. In geometry, however, his work was based on analytic geometry.

In 1859, Cayley was the first person to realize that Euclidean geometry was a special case of projective geometry and ten years later, his projective metric helped people understand the relationship between the different types of non-Euclidean geometries.

In 1872 he was made an honorary fellow of Trinity College, and three years later an ordinary fellow, which meant stipend as well as honor. About this time his friends subscribed for a presentation portrait. Maxwell wrote an address to the committee of subscribers who had charge of the Cayley portrait fund. The verses refer to the subjects investigated in several of Cayley’s most elaborate memoirs; such as, Chapters on the Analytical Geometry of dimensions; On the theory of Determinants; Memoir on the theory of Matrices; Memoirs on skew surfaces, otherwise Scrolls; On the delineation of a Cubic Scroll, etc.[2, p. 17].

In 1876 he published a Treatise on Elliptic Functions, which was his only book. He took great interest in the movement for the University education of women. At Cambridge the women's colleges are Girton and Newnham. In the early days of Girton College he gave direct help in teaching, and for some years he was chairman of the council of Newnham College, in the progress of which he took the keenest interest to the last.

In 1889 the Cambridge University Press requested him to prepare his mathematical papers for publication in a collected form – a request which he appreciated very much. They are printed in magnificent quarto volumes, of which seven appeared under his own editorship. While editing these volumes, he was suffering from a painful internal malady, to which he succumbed on 26 January 1895, in the 74th year of his age. When the funeral took place, a great assemblage met in Trinity Chapel, comprising members of the University, official representatives of Russia and America, and many of the most illustrious philosophers of Britain.

The remainders of his papers were edited by Andrew Forsyth, his successor in the Adlerian Chair. The Collected Mathematical papers number thirteen quarto volumes, and contain 967 papers. Cayley retained to the last his fondness for novel-reading and for travelling. He also took special pleasure in paintings and architecture, and he practiced water-colour painting, which he found useful sometimes in making mathematical diagrams.

1. 
   Arthur Cayley Biography [Electronic resource]. – Mode of access: http://www.thefamouspeople.com/profiles/arthurcayley512.php#bhQ4Y9RPiodVM8IX.99. – Date of access: 15.03.2015.
   
2. 
   Macfarlane, A. Lectures on ten British mathematicians of the nineteenth century / A. Macfarlane. – New York : John Wiley & Sons, 1916. – 164 р.
   

Cтатья посвящена известному английскому математику, президенту Кембриджского философского общества, Лондонского математического общества, Британской ассоциации содействия развитию науки и Королевского астрономического общества Артуру Кэли. Он ввел общепринятое ныне обозначение для определителя, дифференциальных уравнений, эллиптических функций, занимался также сферической астрономией и астрофизикой. Кэли был первым, кто сформулировал определение группы в том виде, как она определяется сегодня – множество с бинарной операцией, удовлетворяющей определённым законам. Прежде же, когда математики говорили о группе, они подразумевали группу перестановок.