| Arthur Cayley | |
 Portrait in London by
Barraud & Jerrard |
|
| Born | August 16, 1821 Richmond, Surrey, UK |
|---|---|
| Died | January 26, 1895 (aged 73) Cambridge, England |
| Residence | England |
| Nationality | British |
| Fields | Mathematician |
| Institutions | University of Cambridge |
| Alma mater | King's College School Trinity College, Cambridge |
| Doctoral advisor | George Peacock William Hopkins |
| Doctoral students | H. F. Baker Andrew Forsyth Charlotte Scott |
| Known for | Projective geometry Group theory |
| Notable awards | Copley Medal (1882) |
Arthur Cayley (August 16, 1821 - January 26, 1895) was a British mathematician. Events 1384 - The Hongwu Emperor of Ming China, Emperor Dong hears a case of a couple who tore paper money bills while fighting Year 1821 ( MDCCCXXI) was a Common year starting on Monday (link will display the full calendar of the Gregorian Calendar (or a Common year Richmond is a town and the principal settlement of the London Borough of Richmond upon Thames in England. Surrey is a county in the South East of England and is one of the Home Counties. The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom, the UK or Britain,is a Sovereign state located Events 1340 - King Edward III of England is declared King of France. Year 1895 ( MDCCCXCV) was a Common year starting on Tuesday (link will display full calendar of the Gregorian calendar (or a Common year The city of Cambridge (ˈkeɪmbrɪdʒ is a university town and the administrative centre of the county of Cambridgeshire, England England is a Country which is part of the United Kingdom. Its inhabitants account for more than 83% of the total UK population whilst its mainland England is a Country which is part of the United Kingdom. Its inhabitants account for more than 83% of the total UK population whilst its mainland England is a Country which is part of the United Kingdom. Its inhabitants account for more than 83% of the total UK population whilst its mainland A mathematician is a person whose primary area of study and research is the field of Mathematics. The University of Cambridge (often Cambridge University) located in Cambridge, England, is the second-oldest university in the Alma mater is Latin for "nourishing mother" It was used in Ancient Rome as a title for the mother Goddess, and in Medieval King's College School Wimbledon or KCS, is one of Britain 's leading independent boys' schools in Wimbledon Trinity College is a constituent college of the University of Cambridge in Cambridge, England. A doctorate is an Academic degree that indicates the highest level of academic achievement George Peacock ( April 9, 1791 – November 8, 1858) was an English Mathematician. William Hopkins FRS (2 February 1793 &ndash 13 October 1866 was an English Mathematician and Geologist. Henry Frederick Baker ( July 3, 1866 - March 17, 1956) was a British mathematician working mainly in Algebraic geometry Andrew Russell Forsyth ( 18 June 1858, Glasgow &ndash 2 June 1942 South Kensington was a Scottish mathematician Charlotte Angas Scott DSc(Lond ( June 8 1858 &ndash November 10 1931, Cambridge, England) was a British Projective geometry is a non- metrical form of Geometry, notable for its principle of duality. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. The Copley Medal is a scientific award for distinguished achievement in any field of Science established by the Royal Society of London in 1731 Events 1384 - The Hongwu Emperor of Ming China, Emperor Dong hears a case of a couple who tore paper money bills while fighting Year 1821 ( MDCCCXXI) was a Common year starting on Monday (link will display the full calendar of the Gregorian Calendar (or a Common year Events 1340 - King Edward III of England is declared King of France. Year 1895 ( MDCCCXCV) was a Common year starting on Tuesday (link will display full calendar of the Gregorian calendar (or a Common year A mathematician is a person whose primary area of study and research is the field of Mathematics. He helped found the modern British school of pure mathematics. Broadly speaking pure mathematics is Mathematics motivated entirely for reasons other than application
As a child, Cayley enjoyed solving complex maths problems for amusement. He entered Trinity College, Cambridge, where he excelled in Greek, French, German, and Italian, as well as mathematics. Trinity College is a constituent college of the University of Cambridge in Cambridge, England. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and He worked as a lawyer for 14 years. A lawyer, according to Black's Law Dictionary, is "a person learned in the law as an attorney, Counsel or Solicitor; a person
He was consequently able to prove the Cayley-Hamilton theorem -- that every square matrix is a root of its own characteristic polynomial. In Linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Hamilton) states that every Square matrix He was the first to define the concept of a group in the modern way -- as a set with a binary operation satisfying certain laws. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element Formerly, when mathematicians spoke of "groups", they had meant permutation groups. In Mathematics, a permutation group is a group G whose elements are Permutations of a given set M, and whose group operation
See also Cayley's theorem. In Group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a Subgroup
Contents |
Early years
Arthur Cayley was born in Richmond, London, England, on August 16 1821. Richmond is a town and the principal settlement of the London Borough of Richmond upon Thames in England. England is a Country which is part of the United Kingdom. Its inhabitants account for more than 83% of the total UK population whilst its mainland His father, Henry Cayley, was a distant cousin of Sir George Cayley the aeronautics innovator, and descended from an ancient Yorkshire family. Sir George Cayley 6th Baronet ( December 27, 1773 &ndash December 15, 1857) sometimes known as "the father of Aerodynamics" was Aeronautics (from Greek aero which means air or sky and nautis which means sailor i Yorkshire is a historic county of Northern England and the largest in Great Britain. He settled in Saint Petersburg, Russia, as a merchant. Saint Petersburg ( tr: Sankt-Peterburg,) is a city and a federal subject of Russia located on the Neva River Russia (Россия Rossiya) or the Russian Federation ( Rossiyskaya Federatsiya) is a transcontinental Country extending Merchants function as professionals who deal with Trade, dealing in commodities that they do not produce themselves in order to produce Profit. His mother was Maria Antonia Doughty, daughter of William Doughty. "Mom" "Mum" and "Mommy" redirect here William Doughty was a United States naval architect who designed many of the sailing 74s. According to some writers she was Russian, but her father's name indicates an English origin. His brother was the linguist Charles Bagot Cayley. Charles Bagot Cayley (1823 - 1883 was a linguist best known for translating Dante into the metre of the original with annotations besides metrical versions of the Arthur spent his first eight years in Saint Petersburg. In 1829 his parents settled permanently at Blackheath, near London. For the game see 1829 (board game. Year 1829 ( MDCCCXXIX) was a Common year starting on Thursday (link will display Blackheath is an area in southeast London centred around a section of open public grassland ('the Heath' and straddling the boundary of the London Borough of Lewisham and the Arthur was sent to a private school. He early showed great liking for, and aptitude in, numerical calculation. At age 14 he was sent to King's College School. King's College School Wimbledon or KCS, is one of Britain 's leading independent boys' schools in Wimbledon The school's master observed indications of mathematical genius and advised the father to educate his son not for his own business, as he had intended, but to enter the University of Cambridge. The University of Cambridge (often Cambridge University) located in Cambridge, England, is the second-oldest university in the
Education
At the unusually early age of 17 Cayley began residence at Trinity College, Cambridge. Trinity College is a constituent college of the University of Cambridge in Cambridge, England. The cause of the Analytical Society had now triumphed, and the Cambridge Mathematical Journal had been instituted by Gregory and Leslie Ellis. The Analytical Society was a group of individuals in early-19th century Britain whose aim was to promote the use of Leibnizian or analytical calculus as opposed to To this journal, at the age of twenty, Cayley contributed three papers, on subjects which had been suggested by reading the Mécanique analytique of Lagrange and some of the works of Laplace.
Cayley finished his undergraduate course by winning the place of Senior Wrangler, and the first Smith's prize. At the University of Cambridge, a Wrangler is a student who has completed the third year (called Part II) of the Mathematical Tripos with First-class The Smith's Prize was the name of each of two prizes awarded annually awarded to two research students in Theoretical Physics, mathematics and Applied mathematics His next step was to take the M. A. degree, and win a Fellowship by competitive examination. His tutor at Cambridge was George Peacock and his private coach was William Hopkins. George Peacock ( April 9, 1791 – November 8, 1858) was an English Mathematician. William Hopkins FRS (2 February 1793 &ndash 13 October 1866 was an English Mathematician and Geologist. He continued to reside at Cambridge for four years; during which time he took some pupils, but his main work was the preparation of 28 memoirs to the Mathematical Journal.
As a lawyer
Because of the limited tenure of his fellowship it was necessary to choose a profession; like De Morgan, Cayley chose law, and at age 25 entered at Lincoln's Inn, London. Augustus De Morgan ( 27 June, 1806 &ndash 18 March, 1871) was a British Mathematician and Logician. The Honourable Society of Lincoln's Inn is one of four Inns of Court in London to which Barristers of England and Wales belong and where He made a specialty of conveyancing. In law conveyancing is the transfer of title of Property from one person to another or the granting of an Encumbrance such as a Mortgage or It was while he was a pupil at the bar examination that he went to Dublin to hear Hamilton's lectures on quaternions. A bar examination is an examination to determine whether a candidate is qualified to practice Law in a given Jurisdiction. Dublin (ˈdʌblɨn/ /ˈdʊblɨn or /ˈdʊbəlɪn/, bˠalʲə aːha klʲiəh or cliə(ɸ is both the largest city and capital of Ireland. Sir William Rowan Hamilton (4 August 1805 &ndash 2 September 1865 was an Irish Mathematician, Physicist, and Astronomer who Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician
His friend Sylvester, his senior by five years at Cambridge, was then an actuary, resident in London; they used to walk together round the courts of Lincoln's Inn, discussing the theory of invariants and covariants. James Joseph Sylvester ( September 3, 1814 London – March 15, 1897 Oxford) was an English Mathematician An actuary is a business professional who deals with the financial impact of risk and uncertainty Invariant theory is a branch of Abstract algebra that studies actions of groups on algebraic varieties from the point of view of their effect During this period of his life, extending over fourteen years, Cayley produced between two and three hundred papers.
As professor
At Cambridge University the ancient professorship of pure mathematics is denominated the Lucasian, and is the chair which had been occupied by Isaac Newton. The incumbent of the Lucasian Chair of Mathematics, the Lucasian Professor is the holder of a mathematical Professorship at the University of Cambridge Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements Around 1860, certain funds bequeathed by Lady Sadleir to the University, having become useless for their original purpose, were employed to establish another professorship of pure mathematics, called the Sadlerian. The Sadleirian Chair is a Professorship in Pure mathematics at the University of Cambridge. The duties of the new professor were defined to be "to explain and teach the principles of pure mathematics and to apply himself to the advancement of that science. " To this chair Cayley was elected when 42 years old. He gave up a lucrative practice for a modest salary; but he never regretted the exchange, for the chair at Cambridge enabled him to end the divided allegiance between law and mathematics, and to devote his energies to the pursuit which he liked best. He at once married and settled down in Cambridge. More fortunate than Hamilton in his choice, his home life was one of great happiness. His friend and fellow investigator, Sylvester, once remarked that Cayley had been much more fortunate than himself; that they both lived as bachelors in London, but that Cayley had married and settled down to a quiet and peaceful life at Cambridge; whereas he had never married, and had been fighting the world all his days.
At first the teaching duty of the Sadlerian professorship was limited to a course of lectures extending over one of the terms of the academic year; but when the University was reformed about 1886, and part of the college funds applied to the better endowment of the University professors, the lectures were extended over two terms. For many years the attendance was small, and came almost entirely from those who had finished their career of preparation for competitive examinations; after the reform the attendance numbered about fifteen. The subject lectured on was generally that of the memoir on which the professor was for the time engaged.
The other duty of the chair - the advancement of mathematical science - was discharged in a handsome manner by the long series of memoirs which he published, ranging over every department of pure mathematics. But it was also discharged in a much less obtrusive way; he became the standing referee on the merits of mathematical papers to many societies both at home and abroad.
In 1876 he published a Treatise on Elliptic Functions, which was his only book. In Complex analysis, an elliptic function is a function defined on the Complex plane which is periodic in two directions (a Doubly-periodic 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. Girton College is one of the constituent colleges of the University of Cambridge in Cambridge, England. Newnham College is a Women's college in the University of Cambridge.
In 1872 he was made an honorary fellow of Trinity College, and three years later an ordinary fellow, which meant stipend as well as honour. 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. James Clerk Maxwell (13 June 1831 &ndash 5 November 1879 was a Scottish mathematician and theoretical physicist. The verses refer to the subjects investigated in several of Cayley's most elaborate memoirs; such as, Chapters on the Analytical Geometry of n 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. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n
In 1881 he received from the Johns Hopkins University, Baltimore, where Sylvester was then professor of mathematics, an invitation to deliver a course of lectures. He accepted the invitation, and lectured at Baltimore during the first five months of 1882 on the subject of the Abelian and Theta Functions.
BMA
The next year Cayley came prominently before the world, as President of the British Association for the Advancement of Science. The meeting was held at Southport, in the north of England. As the President's address is one of the great popular events of the meeting, and brings out an audience of general culture, it is usually made as little technical as possible. Hamilton was the kind of mathematician to suit such an occasion, but he never got the office, on account of his occasional breaks. Cayley had not the oratorical, the philosophical, or the poetical gifts of Hamilton, but then he was an eminently safe man. He took for his subject the Progress of Pure Mathematics; and he opened his address in the following naive manner:
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I wish to speak to you to-night upon Mathematics. I am quite aware of the difficulty arising from the abstract nature of my subject; and if, as I fear, many or some of you, recalling the providential addresses at former meetings, should wish that you were now about to have from a different President a discourse on a different subject, I can very well sympathize with you in the feeling. But be that as it may, I think it is more respectful to you that I should speak to you upon and do my best to interest you in the subject which has occupied me, and in which I am myself most interested. And in another point of view, I think it is right that the address of a president should be on his own subject, and that different subjects should be thus brought in turn before the meetings. So much the worse, it may be, for a particular meeting: but the meeting is the individual, which on evolution principles, must be sacrificed for the development of the race.
Cayley doubtless felt that he was addressing not only the popular audience then and there before him, but the mathematicians of distant places and future times; for the address is a valuable historical review of various mathematical theories, and is characterized by freshness, independence of view, suggestiveness, and learning.
The Collected Papers
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 January 26, 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 United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom, the UK or Britain,is a Sovereign state located
The remainder of his papers were edited by Prof. Forsyth, his successor in the Sadlerian chair. The Collected Mathematical papers number thirteen quarto volumes, and contain 967 papers. His writings are his best monument, and certainly no mathematician has ever had his monument in grander style. De Morgan's works would be more extensive, and much more useful, but he did not have behind him a University Press. As regards fads, 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-color painting, which he found useful sometimes in making mathematical diagrams.
Quaternions
To the third edition of P. G. Tait's Elementary Treatise on Quaternions, Cayley contributed a chapter entitled "Sketch of the analytical theory of quaternions. Peter Guthrie Tait ( April 28, 1831 - July 4, 1901) was a Scottish mathematical physicist, best known for the seminal energy Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician " In it the √−1 reappears in all its glory, and in entire, so it is said, independence of i, j, k.
In 1894 there arose a brisk discussion between Tait and Cayley on "Coordinates versus Quaternions," the record of which is printed in the Proceedings of the Royal Society of Edinburgh. The Royal Society of Edinburgh is Scotland 's National academy of science and letters Cayley maintained the position that while coordinates are applicable to the whole science of geometry and are the natural and appropriate basis and method in the science, quaternions seemed a particular and very artificial method for treating such parts of the science of three-dimensional geometry as are most naturally discussed by means of the rectangular coordinates x, y, z. In the course of his paper Cayley says:
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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. As another illustration, I compare a quaternion formula to a pocket-map—a capital thing to put in one's pocket, but which for use must be unfolded: the formula, to be understood, must be translated into coordinates.
He goes on to say,
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I remark that the imaginary of ordinary algebra—for distinction call this θ—has no relation whatever to the quaternion symbols i, j, k; in fact, in the general point of view, all the quantities which present themselves, are, or may be, complex values a + θb, or in other words, say that a scalar quantity is in general of the form a + θb. Thus quaternions do not properly present themselves in plane or two-dimensional geometry at all; but they belong essentially to solid or three-dimensional geometry, and they are most naturally applicable to the class of problems which in coordinates are dealt with by means of the three rectangular coordinates x, y, z.
To the pocketbook illustration it may be replied that a set of coordinates is an immense wall map, which you cannot carry about, even though you should roll it up, and therefore is useless for many important purposes. In reply to the arguments, it may be said, first, √−1 has a relation to the symbols i, j, k for each of these can be analyzed into a unit axis multiplied by √−1; second, as regards plane geometry, the ordinary form of complex quantity is a degraded form of the quaternion in which the constant axis of the plane is left unspecified. In Mathematics, plane geometry may mean geometry of a plane, geometry of the Euclidean plane, or sometimes Cayley took his illustrations from his experience as a traveller. Tait brought forward an illustration from which you might imagine he had visited the Bethlehem Iron Works, and hunted tigers in India. He says,
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A much more natural and adequate comparison would, it seems to me, liken Coordinate Geometry to a steam-hammer, which an expert may employ on any destructive or constructive work of one general kind, say the cracking of an eggshell, or the welding of an anchor. But you must have your expert to manage it, for without him it is useless. He has to toil amid the heat, smoke, grime, grease, and perpetual din of the suffocating engine-room. The work has to be brought to the hammer, for it cannot usually be taken to its work. And it is not in general, transferable; for each expert, as a rule, knows, fully and confidently, the working details of his own weapon only. Quaternions, on the other hand, are like the elephant's trunk, ready at any moment for anything, be it to pick up a crumb or a field-gun, to strangle a tiger, or uproot a tree; portable in the extreme, applicable anywhere—like in the trackless jungle and in the barrack square—directed by a little native who requires no special skill or training, and who can be transferred from one elephant to another without much hesitation. Surely this, which adapts itself to its work, is the grander instrument. But then, it is the natural, the other, the artificial one.
The reply which Tait makes, so far as it is an argument, is: There are two systems of quaternions, the i, j, k one, and another one which Hamilton developed from it; Cayley knows the first only, he himself knows the second; the former is an intensely artificial system of imaginaries, the latter is the natural organ of expression for quantities in space. Should a fourth edition of his Elementary Treatise be called for i, j, k will disappear from it, excepting in Cayley's chapter, should it be retained. Tait thus describes the first system:
- Hamilton's extraordinary Preface to his first great book shows how from Double Algebras, through Triplets, Triads, and Sets, he finally reached Quaternions. This was the genesis of the Quaternions of the forties, and the creature thus produced is still essentially the Quaternion of Prof. Cayley. It is a magnificent analytical conception; but it is nothing more than the full development of the system of imaginaries i, j, k; defined by the equations, i² = j² = k² = ijk = −1 with the associative, but not the commutative, law for the factors. The novel and splendid points in it were the treatment of all directions in space as essentially alike in character, and the recognition of the unit vector's claim to rank also as a quadrantal versor. These were indeed inventions of the first magnitude, and of vast importance. And here I thoroughly agree with Prof. Cayley in his admiration. Considered as an analytical system, based throughout on pure imaginaries, the Quaternion method is elegant in the extreme. But, unless it had been also something more, something very different and much higher in the scale of development, I should have been content to admire it;—and to pass it by.
From "the most intensely artificial of systems, arose, as if by magic, an absolutely natural one" which Tait thus further describes. "To me Quaternions are primarily a Mode of Representation:—immensely superior to, but of essentially the same kind of usefulness as, a diagram or a model. They are, virtually, the thing represented; and are thus antecedent to, and independent of, coordinates; giving, in general, all the main relations, in the problem to which they are applied, without the necessity of appealing to coordinates at all. Coordinates may, however, easily be read into them:—when anything (such as metrical or numerical detail) is to be gained thereby. Quaternions, in a word, exist in space, and we have only to recognize them:—but we have to invent or imagine coordinates of all kinds. "
To meet the objection why Hamilton did not throw i, j, k overboard, and expound the developed system, Tait says:
- Most unfortunately, alike for himself and for his grand conception, Hamilton's nerve failed him in the composition of his first great volume. Had he then renounced, for ever, all dealings with i, j, k, his triumph would have been complete. He spared Agog, and the best of the sheep, and did not utterly destroy them. He had a paternal fondness for i, j, k; perhaps also a not unnatural liking for a meretricious title such as the mysterious word Quaternion; and, above all, he had an earnest desire to make the utmost return in his power for the liberality shown him by the authorities of Trinity College, Dublin. He had fully recognized, and proved to others, that his i, j, k, were mere excrescences and blots on his improved method:---but he unfortunately considered that their continued (if only partial) recognition was indispensable to the reception of his method by a world steeped in—Cartesianism! Through the whole compass of each of his tremendous volumes one can find traces of his desire to avoid even an allusion to i, j, k, and along with them, his sorrowful conviction that, should he do so, he would be left without a single reader.
Philosophy
To Cayley's presidential address we are indebted for information about the view which he took of the foundations of exact science, and the philosophy which commended itself to his mind. He quoted Plato and Kant with approval, J. S. Mill with faint praise. Biography Early life Birth and family Plato was born in Athens Greece Immanuel Kant (ɪmanuəl kant 22 April 1724 12 February 1804 was an 18th-century German Philosopher from the Prussian city of Königsberg John Stuart Mill (20 May 1806 &ndash 8 May 1873 British Philosopher, political economist, civil servant and Member of Parliament, was an influential Although he threw a sop to the empirical philosophers at the beginning of his address, he gave them something to think of before he finished.
He first of all remarks that the connection of arithmetic and algebra with the notion of time is far less obvious than that of geometry with the notion of space; in which he, of course, made a hit at Hamilton's theory of Algebra as the science of pure time. Further on he discusses the theory directly, and concludes as follows:
- Hamilton uses the term algebra in a very wide sense, but whatever else he includes under it, he includes all that in contradistinction to the Differential Calculus would be called algebra. Differential Calculus, a field in Mathematics, is the study of how functions change when their inputs change Using the word in this restricted sense, I cannot myself recognize the connection of algebra with the notion of time; granting that the notion of continuous progression presents itself and is of importance, I do not see that it is in anywise the fundamental notion of the science. And still less can I appreciate the manner in which the author connects with the notion of time his algebraic couple, or imaginary magnitude, a + b√−1.
So you will observe that doctors differ—Tait and Cayley—about the soundness of Hamilton's theory of couples. But it can be shown that a couple may not only be represented on a straight line, but actually means a portion of a straight line; and as a line is unidimensional, this favors the truth of Hamilton's theory.
As to the nature of mathematical science Cayley quoted with approval from an address of Hamilton's:
- These purely mathematical sciences of algebra and geometry are sciences of the pure reason, deriving no weight and no assistance from experiment, and isolated or at least isolable from all outward and accidental phenomena. The idea of order with its subordinate ideas of number and figure, we must not call innate ideas, if that phrase be defined to imply that all men must possess them with equal clearness and fulness; they are, however, ideas which seem to be so far born with us that the possession of them in any conceivable degree is only the development of our original powers, the unfolding of our proper humanity.
It is the aim of the evolution philosopher to reduce all knowledge to the empirical status; the only intuition he grants is a kind of instinct formed by the experience of ancestors and transmitted cumulatively by heredity. Cayley first takes him up on the subject of arithmetic:
- Whatever difficulty be raisable as to geometry, it seems to me that no similar difficulty applies to arithmetic; mathematician, or not, we have each of us, in its most abstract form, the idea of number; we can each of us appreciate the truth of a proposition in numbers; and we cannot but see that a truth in regard to numbers is something different in kind from an experimental truth generalized from experience. Compare, for instance, the proposition, that the sun, having already risen so many times, will rise to-morrow, and the next day, and the day after that, and so on; and the proposition that even and odd numbers succeed each other alternately ad infinitum; the latter at least seems to have the characters of universality and necessity. Or again, suppose a proposition observed to hold good for a long series of numbers, one thousand numbers, two thousand numbers, as the case may be: this is not only no proof, but it is absolutely no evidence, that the proposition is a true proposition, holding good for all numbers whatever; there are in the Theory of Numbers very remarkable instances of propositions observed to hold good for very long series of numbers which are nevertheless untrue.
Then he takes him up on the subject of geometry, where the empiricist rather boasts of his success.
- It is well known that Euclid's fifth axiom, even in Playfair's form of it, has been considered as needing demonstration; and that Lobatschewsky constructed a perfectly consistent theory, wherein this axiom was assumed not to hold good, or say a system of non-Euclidean plane geometry. In Geometry, the parallel postulate, also called Euclid 's fifth postulate since it is the fifth postulate in Euclid's ''Elements'', is a distinctive Professor John Playfair FRSE ( March 10, 1748 &ndash July 20, 1819) was a Scottish Scientist. Nikolai Ivanovich Lobachevsky (Никола́й Ива́нович Лобаче́вский ( December 1 1792 &ndash February 24 1856 ( N In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry My own view is that Euclid's fifth axiom in Playfair's form of it does not need demonstration, but is part of our notion of space, of the physical space of our experience---the space, that is, which we become acquainted with by experience, but which is the representation lying at the foundation of all external experience. Riemann's view before referred to may I think be said to be that, having in intellectu a more general notion of space (in fact a notion of non-Euclidean space), we learn by experience that space (the physical space of our experience) is, if not exactly, at least to the highest degree of approximation, Euclidean space. But suppose the physical space of our experience to be thus only approximately Euclidean space, what is the consequence which follows? Not that the propositions of geometry are only approximately true, but that they remain absolutely true in regard to that Euclidean space which has been so long regarded as being the physical space of our experience.
In his address he remarks that the fundamental notion which underlies and pervades the whole of modern analysis and geometry is that of imaginary magnitude in analysis and of imaginary space (or space as a locus in quo of imaginary points and figures) in geometry. In the case of two given curves there are two equations satisfied by the coordinates (x, y) of the several points of intersection, and these give rise to an equation of a certain order for the coordinate x or y of a point of intersection. In the case of a straight line and a circle this is a quadratic equation; it has two roots real or imaginary. There are thus two values, say of x, and to each of these corresponds a single value of y. There are therefore two points of intersection, viz. , a straight line and a circle intersect always in two points, real or imaginary. It is in this way we are led analytically to the notion of imaginary points in geometry. He asks, What is an imaginary point? Is there in a plane a point the coordinates of which have given imaginary values? He seems to say No, and to fall back on the notion of an imaginary space as the locus in quo of the imaginary point.
List of notions named for Arthur Cayley
Also named after Arthur Cayley
- Cayley crater on the Moon
- Cayley mathematics contest created by the University of Waterloo
Works by Arthur Cayley
- An element treatise on elliptic functions (Cambridge : Deighton : Bell, 1876)
- The collected mathematical papers of Arthur Cayley (Volume 1) (Cambridge, University Press, 1889-1897)
- The collected mathematical papers of Arthur Cayley (Volume 2) (Cambridge, University Press, 1889-1897)
- The collected mathematical papers of Arthur Cayley (Volume 3) (Cambridge, University Press, 1889-1897)
- The collected mathematical papers of Arthur Cayley (Volume 4) (Cambridge, University Press, 1889-1897)
- The collected mathematical papers of Arthur Cayley (Volume 5) (Cambridge, University Press, 1889-1897)
- The collected mathematical papers of Arthur Cayley (Volume 6) (Cambridge, University Press, 1889-1897)
- The collected mathematical papers of Arthur Cayley (Volume 7) (Cambridge, University Press, 1889-1897)
- The collected mathematical papers of Arthur Cayley (Volume 8) (Cambridge, University Press, 1889-1897)
- The collected mathematical papers of Arthur Cayley (Volume 9) (Cambridge, University Press, 1889-1897)
- The collected mathematical papers of Arthur Cayley (Volume 10) (Cambridge, University Press, 1889-1897)
- The collected mathematical papers of Arthur Cayley (Volume 11) (Cambridge, University Press, 1889-1897)
- The collected mathematical papers of Arthur Cayley (Volume 12) (Cambridge, University Press, 1889-1897)
- The collected mathematical papers of Arthur Cayley (Volume 13) (Cambridge, University Press, 1889-1897)
Bibliography
Primary:
- 1883. In Group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a Subgroup In Linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Hamilton) states that every Square matrix Linear algebra is the branch of Mathematics concerned with Grassmann–Cayley algebra is a form of modelling Algebra for Projective geometry, based on work by German mathematician Hermann Grassmann on Exterior Distance geometry is the characterization and study of sets of points based only on given values of the Distances between member pairs In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real In Mathematics, the Cayley graph, also known as the Cayley colour graph, is the graph that encodes the structure of a Discrete group. A Cayley table, after the 19th century British Mathematician Arthur Cayley, describes the structure of a The Cayley-Purser algorithm was a Public-key cryptography Algorithm published in early 1999 by 16-year-old Irishwoman Sarah Flannery In Mathematics, Cayley's formula is a result in Graph theory named after Arthur Cayley. In geometry the Klein model, also called the projective model the Beltrami–Klein model the Klein–Beltrami model and the Cayley–Klein model is a model of n-dimensional Hyperbolic In In Mathematics, the Cayley transform, named after Arthur Cayley, has a cluster of related meanings Cayley is a small lunar Impact crater that is located in a Basaltic Lava -flooded region to the west of Mare Tranquillitatis. The University of Waterloo (also referred to as UW and Waterloo) is a comprehensive Public university in the city of Waterloo, Ontario "Presidential address to the British Association" in Ewald, William B. , ed. , 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford Uni. Press: 542-73.
Secondary:
- Lectures on Ten British Mathematicians of the Nineteenth Century by Alexander MacFarlane (complete text at Project Gutenberg)
- T. Alexander Macfarlane ( April 21 1851 – August 28, 1913) was a Scottish - Canadian Logician Physicist Project Gutenberg, abbreviated as PG, is a volunteer effort to Digitize, archive and distribute Cultural works Crilly, "A Victorian mathematician: Arthur Cayley (1821-1895)," The Mathematical Gazette, Vol. 79, No. 485, 1995, pp. 259-262.
External links
- O'Connor, John J. & Robertson, Edmund F. , “Arthur Cayley”, MacTutor History of Mathematics archive
- Arthur Cayley at the Mathematics Genealogy Project
- Weisstein, Eric W. The MacTutor History of Mathematics archive is an award-winning website maintained by John J The Mathematics Genealogy Project is a web-based Database that gives an Academic genealogy based on Dissertation supervision relations , Cayley, Arthur (1821-1895) at ScienceWorld. ScienceWorld, also known as Eric Weisstein's World of Science, is a Web site that opened to the general public in January 2002.
| Persondata | |
|---|---|
| NAME | Cayley, Arthur |
| ALTERNATIVE NAMES | |
| SHORT DESCRIPTION | British mathematician |
| DATE OF BIRTH | August 16, 1821 |
| PLACE OF BIRTH | Richmond, Surrey, UK |
| DATE OF DEATH | January 26, 1895 |
| PLACE OF DEATH | Cambridge, UK |
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