- Born:
- August 16, 1821, Richmond, Surrey, England
- Died:
- January 26, 1895, Cambridge, Cambridgeshire (aged 73)
- Awards And Honors:
- Copley Medal (1882)
- Subjects Of Study:
- invariant
- matrix
- multidimensional space
Arthur Cayley
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Who was Arthur Cayley?
Arthur Cayley (1821â1895) was a leading English mathematician and a central figure in the British school of pure mathematics. He was educated at Trinity College, Cambridge, and later pursued law, practicing in London from 1849 to 1863 while writing over 300 mathematical papers.
In 1863, Cayley became the Sadleirian Professor of Mathematics at Cambridge, dedicating himself fully to mathematical research. His work spanned several areas, including algebraic theory of curves and surfaces, group theory, linear algebra, and elliptic functions. Cayley formalized matrix theory and contributed significantly to invariant theory, studying properties of forms unchanged under transformations. In 1854, he was the first to define a group abstractly.
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What subjects did Arthur Cayley study and work in?
Arthur Cayley studied law and mathematics. After graduating as Senior Wrangler from Cambridge in 1842, he remained there until 1846. He then became a lawyer, practicing from 1849 to 1863, while also writing mathematical papers. In 1863, he became a professor of mathematics at Cambridge. Cayley made significant contributions to the algebraic theory of curves and surfaces, group theory, linear algebra, graph theory, combinatorics, and elliptic functions. He also worked in mechanics and astronomy, with a focus on lunar studies.
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What are some of Arthur Cayleyâs most important mathematical discoveries?
Arthur Cayley was a leading force in the British school of pure mathematics during the 19th century. He made significant contributions to various areas of mathematics, including algebra, geometry, group theory, and elliptic functions.
Cayley formalized matrix theory, introducing matrices as autonomous mathematical objects with specific rules for manipulation. His work on quantics, also known as algebraic forms, led to the establishment of invariant theory, which studies properties unchanged under transformations. In geometry, Cayley developed a notion of distance in projective geometry and realized Euclidean geometry as a special case of projective geometry.
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How did Arthur Cayley contribute to the study of algebra?
Arthur Cayley was a leading force in the British school of pure mathematics during the 19th century. His contributions spanned several areas, notably the algebraic theory of curves and surfaces, group theory, and linear algebra.
Cayley formalized matrix theory, conceiving matrices as autonomous mathematical objects with special rules for manipulation. He demonstrated how to derive the inverse matrix using the determinant of the original matrix. His work, along with James Joseph Sylvester, led to the establishment of invariant theory, focusing on properties of forms unchanged under transformations. Cayley's approach to algebra, influenced by the British tradition of symbolic algebra, shifted the focus from studying objects to operations among abstract objects, broadening the scope of algebraic thinking.
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What is the Cayley table, and why is it important in mathematics?
A Cayley table defines a group by presenting all possible combinations of its elements under the group's operation. Arthur Cayley was the first to explicitly define a group abstractly in 1854, using a set of meaningless symbols with an operation defined on them. He specified the properties of the operation without referring to the nature of the elements.
Cayley's work had little immediate impact, but he correctly deduced some basic properties of the group, such as that if the group has n elements, then (\theta^n = 1) for each element (\theta). Arthur Cayley made important contributions to group theory, linear algebra, and combinatorics.
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How has Arthur Cayleyâs work influenced modern mathematics?
Arthur Cayley, a leading 19th-century British mathematician, significantly shaped modern mathematics through his work in various areas.
Cayley formalized matrix theory, which became a fundamental tool in analysis, geometry, and especially in the emerging discipline of linear algebra. His work provided a new instance of a system with well-elaborated arithmetic, whose rules departed from traditional number systems in that multiplication was not generally commutative.
In 1854, Cayley was the first to realize that a group could be defined abstractly by specifying the properties of the operation defined on its elements. Cayley made important contributions to the algebraic theory of curves and surfaces, group theory, linear algebra, graph theory, combinatorics, and elliptic functions.
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Arthur Cayley (born August 16, 1821, Richmond, Surrey, Englandâdied January 26, 1895, Cambridge, Cambridgeshire) was an English mathematician and leader of the British school of pure mathematics that emerged in the 19th century.
Although Cayley was born in England, his first seven years were spent in St. Petersburg, Russia, where his parents lived in a trading community affiliated with the Muscovy Company. On the familyâs permanent return to England in 1828 he was educated at a small private school in Blackheath, followed by the three-year course at Kingâs College, London. Cayley entered Trinity College, Cambridge, in 1838 and emerged as the champion student of 1842, the âSenior Wranglerâ of his year. A fellowship enabled him to stay on at Cambridge, but in 1846 he left the university to study the law at Lincolnâs Inn in London. Cayley practised law in London from 1849 until 1863, while writing more than 300 mathematical papers in his spare time. In recognition of his mathematical work, he was elected to the Royal Society in 1852 and presented with its Royal Medal seven years later. In 1863 he accepted the Sadleirian professorship in mathematics at Cambridgeâsacrificing his legal career in order to devote himself full-time to mathematical research. In that same year he married Susan Moline, the daughter of a country banker.
Cayleyâs manner was diffident but decisive. He was a capable administrator who quietly and effectively discharged his academic duties. He was an early supporter of womenâs higher education and steered Newnham College, Cambridge (founded in 1871), during the 1880s. Despite aiding the careers of a few students who naturally took to pure mathematics, Cayley never established a full-fledged research school of mathematics at Cambridge.
In mathematics Cayley was an individualist. He handled calculations and symbolic manipulations with formidable skill, guided by a deep intuitive understanding of mathematical theories and their interconnections. His ability to keep abreast of current work while seeing the wider view enabled him to perceive important trends and to make valuable suggestions for further investigation.
Cayley made important contributions to the algebraic theory of curves and surfaces, group theory, linear algebra, graph theory, combinatorics, and elliptic functions. He formalized the theory of matrices. Among Cayleyâs most important papers were his series of 10 âMemoirs on Quanticsâ (1854â78). A quantic, known today as an algebraic form, is a polynomial with the same total degree for each term; for example, every term in the following polynomial has a total degree of 3: x3 + 7x2y â 5xy2 + y3. Alongside work produced by his friend James Joseph Sylvester, Cayleyâs study of various properties of forms that are unchanged (invariant) under some transformation, such as rotating or translating the coordinate axes, established a branch of algebra known as invariant theory.
In geometry Cayley concentrated his attention on analytic geometry, for which he naturally employed invariant theory. For example, he showed that the order of points formed by intersecting lines is always invariant, regardless of any spatial transformation. In 1859 Cayley outlined a notion of distance in projective geometry (a projective metric), and he was one of the first to realize that Euclidean geometry is a special case of projective geometryâan insight that reversed current thinking. Ten years later, Cayleyâs projective metric provided a key for understanding the relationship between the various types of non-Euclidean geometries.
While Cayley was essentially a pure mathematician, he also pursued mechanics and astronomy. He was active in lunar studies and produced two widely praised reports on dynamics (1857, 1862). Cayley had an extraordinarily prolific career, producing almost a thousand mathematical papers. His habit was to embark on long studies punctuated by rapidly written âbulletins from the front.â Cayley wrote French effortlessly and often published in Continental journals. As a young graduate at Cambridge, he was inspired by the work of the mathematician Karl Jacobi (1804â51), and in 1876 Cayley published his only book, An Elementary Treatise on Elliptic Functions, which drew out this widely studied subject from Jacobiâs point of view.
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Cayley was awarded numerous honours, including the Copley Medal in 1882 by the Royal Society. At various times he was president of the Cambridge Philosophical Society, the London Mathematical Society, the British Association for the Advancement of Science, and the Royal Astronomical Society.