De morgan augustus biography of william

In the chair of mental philosophy in University College fell vacant. James Martineaua Unitarian clergyman and professor of mental philosophy, was recommended formally by the Senate to the Council; but in the Council there were some who objected to a Unitarian clergyman, and others who objected to theistic philosophy.

De morgan augustus biography of william: He was born and served in

A layman of the school of Bain and Spencer was appointed. De Morgan considered that the old standard of religious neutrality had been hauled down, and forthwith resigned. He was now 60 years of age. Two years later his son George—the "younger Bernoulli", as Augustus loved to hear him called, in de morgan augustus biography of william to the eminent father-and-son mathematicians of that name—died.

This blow was followed by the death of a daughter. Five years after his resignation from University College De Morgan died of nervous prostration on 18 March De Morgan was a brilliant and witty writer, whether as a controversialist or as a correspondent. In his time there flourished two Sir William Hamiltons who have often been conflated.

One was Sir William Hamilton, 9th Baronet that is, his title was inheriteda Scotsman, professor of logic and metaphysics at the University of Edinburgh ; the other was a knight that is, won the titlean Irishman, professor at astronomy in the University of Dublin. The baronet contributed to logic, especially the doctrine of the quantification of the predicate; the knight, whose full name was William Rowan Hamiltoncontributed to mathematics, especially geometric algebraand first described the Quaternions.

De Morgan was interested in the work of both, and corresponded with both; but the correspondence with the Scotsman ended in a public controversy, whereas that with the Irishman was marked by friendship and terminated only by death. In one of his letters to Rowan, De Morgan says. The correspondence of De Morgan with Hamilton the mathematician extended over twenty-four years; it contains discussions not only of mathematical matters, but also of subjects of general interest.

It is marked by geniality on the part of Hamilton and by wit on the part of De Morgan. The following is a specimen: Hamilton wrote. De Morgan was full of personal peculiarities. On the occasion of the installation of his friend, Lord Brougham, as Rector of the University of Edinburgh, the Senate offered to confer on him the honorary degree of LL.

He disliked the provinces outside London, and while his family enjoyed the seaside, and men of science were having a good time at a meeting of the British Association in the country, he remained in the hot and dusty libraries of the metropolis. He said that he felt like Socrateswho declared that the farther he was from Athens the farther was he from happiness.

He never sought to become a Fellow of the Royal Societyand he never attended a meeting of the Society; he said that he had no ideas or sympathies in common with the physical philosopher. His attitude was possibly due to his physical infirmity, which prevented him from being either an observer or an experimenter. Were the writings of De Morgan, such as his contributions to the Useful Knowledge Society, published in the form of collected works, they would form a small library Mainly through the efforts of Peacock and Whewell, a Philosophical Society had been inaugurated at Cambridge, and De Morgan contributed four memoirs to its transactions on the foundations of algebra, and an equal number on formal logic.

The best presentation of his view of algebra is found in a volume, entitled Trigonometry and Double Algebrapublished in ; and his earlier view of formal logic is found in a volume published in George Peacock's theory of algebra was much improved by D. Gregorya younger member of the Cambridge School, who laid stress not on the permanence of equivalent forms, but on the permanence of certain formal laws.

This new theory of algebra as the science of symbols and of their laws of combination was carried to its logical issue by De Morgan; and his doctrine on the subject is still followed by English algebraists in general. Thus George Chrystal founds his Textbook of Algebra on De Morgan's theory; although an attentive reader may remark that he practically abandons it when he takes up the subject of infinite series.

De Morgan's work entitled Trigonometry and Double Algebra [6] consists of two parts; the former of which is a treatise on trigonometryand the latter a treatise on generalized algebra which he called "double algebra". The next stage is universal arithmeticwhere letters appear instead of numbers, so as to denote numbers universally, and the processes are conducted without knowing the values of the symbols.

Let a and b denote any natural numbers. The third stage is single algebrawhere the symbol may denote a quantity forwards or a quantity backwards, and is adequately represented by segments on a straight line passing through an origin. Negative quantities are then no longer impossible; they are represented by the backward segment. The fourth stage is double algebra.

The algebraic symbol denotes in general a segment of a line in a given plane. The remarkable fact is that this double algebra satisfies all the fundamental laws above enumerated, and as every apparently impossible combination of symbols has been interpreted it looks like the complete form of algebra. In chapter 6 he introduced hyperbolic functions and discussed the connection of common and hyperbolic trigonometry.

De Morgan and many others worked hard at the problem, but nothing came of it until the problem was taken up by Hamilton. We now see the reason clearly: The symbol of double algebra denotes not a length and a direction; but a multiplier and an angle. In it the angles are confined to one plane. Hence the next stage will be a quadruple de morgan augustus biography of williamwhen the axis of the plane is made variable.

And this gives the answer to the first question; double algebra is nothing but analytical plane trigonometry, and this is why it has been found to be the natural analysis for alternating currents. But De Morgan never got this far. In Book II, Chapter II, following the above quoted passage about the theory of symbolic algebra, De Morgan proceeds to give an inventory of the fundamental symbols of algebra, and also an inventory of the laws of algebra.

As De Morgan explains, the last of these symbols represents writing a latter expression in superscript over and after a former. His inventory of the fundamental laws is expressed under fourteen heads, but some of them are merely definitions. The preceding list of symbols is the matter under the first of these heads. De Morgan professes to give a complete inventory of the laws which the symbols of algebra must obey, for he says, "Any system of symbols which obeys these rules and no others—except they be formed by combination of these rules—and which uses the preceding symbols and no others—except they be new symbols invented in abbreviation of combinations of these symbols—is symbolic algebra.

If the commutative law fails, the associative may hold good; but not vice versa. Why does he not give it full scope? De Morgan was an early proponent of symbolical algebra. First expressed by George Peacock in his Treatise on Algebra and developed by Duncan Gregorysymbolical algebra was a first step towards abstract algebraseparating the manipulation of symbols from their arithmetic meaning.

De Morgan would move on from symbolical algebra to develop what he called "logical" or " double " algebra in a series of papers [ 38 ] [ 39 ] [ 40 ] [ 41 ] and the book Trigonometry and Double Algebra De Morgan and Hamilton were friends and correspondents for over 25 years, with De Morgan serving both as a colleague in mathematics, reviewing his Lectures on Quaternionsand as a confidant on personal matters.

The study of logic in Britain underwent a revival following the publication of Richard Whately 's Elements of Logic in The book itself was the subject of a debate that would spur both De Morgan and George Boole to action. On the one hand, argued by William Whewelllogic, particularly syllogism as emphasized by Whately, could not arrive at "new truths" and was therefore inferior to and distinct from scientific reasoning; on the other hand, argued by the Scottish philosopher Sir William HamiltonWhately's effort to equate logic to a "grammar for reasoning" was wrong and reductive.

De Morgan's paper "On the structure of the syllogism", [ 45 ] published inmathematically defines the rules of Aristotelian logicspecifically syllogismand including what are now known as De Morgan's laws. Historically significant as the inception of mathematical logic[ 46 ] at the time, De Morgan's paper initiated a dispute with Hamilton over the role of mathematics in logic; "mathematics can not conduce to logical habits at all," Hamilton would write.

On realizing this, Hamilton would claim that De Morgan had committed plagiarism. Boole's work would eclipse De Morgan's and come to define early mathematical logic. De Morgan continued to support Boole's efforts, proofreading and advocating for Boole's work. Upon Boole's death, De Morgan worked to ensure Boole's family received a government pension.

De Morgan was so struck by the work that he entered into correspondence with Ramchundra and arranged for the book's re-publication in London intargeting a European audience; De Morgan's preface surveyed classical Indian mathematical thought and urged a contemporary return of Indian mathematics: [ 51 ] [ 52 ] [ 28 ]. On examining this work I saw in it, not merely merit worthy of encouragement, but merit of a peculiar kind, the encouragement of which, as it appeared to me, was likely to promote native effort towards the restoration of the native mind in India.

The influence of classical Indian logic on De Morgan's own work on logic has been speculated upon. Think what must have been the effect of the intense Hinduizing of three such men as Babbage, De Morgan, and George Boole on the mathematical atmosphere of — What share had it in generating the vector analysis and the mathematics by which investigations in physical science are now conducted?

Arthur Cowper Ranyard and George Campbell De Morgan, De Morgan's son, conceived the idea of founding a mathematical society in London, where mathematical papers would be not only received as by the Royal Society but also read and discussed. De Morgan was the first president and his son was the first secretary. Augustus was one of seven children, only four of whom survived to adulthood.

Both had studied mathematics at Cambridge and subsequently left for religious reasons, and both were actuaries. De Morgan had three sons and four daughters, including fairytale author Mary De Morgan. His second son, George, acquired distinction in mathematics at University College and the University of London. De Morgan was full of personal peculiarities.

On the occasion of the installation of his friend, Lord Brougham, as Rector of the University of Edinburgh, the Senate offered to confer on him the honorary degree of LL. He humorously described himself using the Latin phrase ' Homo paucarum literarum ' man of few lettersreflecting his modesty about his extensive contributions to mathematics and logic.

De morgan augustus biography of william: Augustus De Morgan was an English

He disliked the provinces outside London, and while his family enjoyed the seaside and men of science were having a good time at a meeting of the British Association in the country, he remained in the hot and dusty libraries of the metropolis. He said that he felt like Socrateswho declared that the farther he was from Athensthe farther he was from happiness.

He never sought to become a Fellow of the Royal Society and he never attended a meeting of the Society. He said that he had no ideas or sympathies in common with the physical philosopher; his attitude was possibly due to his physical infirmity, which prevented him from being either an observer or an experimenter. He never voted at an election, and he never visited the House of Commonsthe Tower of Londonor Westminster Abbey.

Despite a strict Church of England upbringing [ 59 ] De Morgan was publicly a non-conformistat some personal cost: His refusal to conform debarred him from further advancement at Cambridge; his marriage was without Church ceremony; [ 60 ] and on several occasions he fought with the University College administration to maintain religious neutrality, [ 61 ] eventually resigning over the issue.

De Morgan was on occasion accused of atheism [ 65 ] which he dismissed as sectarianism. I commend my future with hope and confidence to Almighty God; to God the Father of our Lord Jesus Christ, whom I believe in my heart to be the Son of God, but whom I have not confessed with my lips, because in my time such confession has always been the way up in the world.

Two years later, his son George—the "younger Bernoulli," as Augustus loved to hear him called, [ 69 ] in allusion to the eminent father-and-son mathematicians of that name—died. This blow was followed by the death of a daughter. Five years after his resignation from University College, De Morgan died of nervous prostration on 18 March De Morgan is best known for his pioneering contributions to mathematical logicspecifically algebraic logicand, to a lesser extent, for his contributions to the beginnings of abstract algebra.

De Morgan's contributions to logic are two-fold. Firstly, before De Morgan there was no mathematical logic— logicincluding formal logicwas the domain of philosophers; De Morgan was the first to make formal logic a mathematical subject. Secondly, De Morgan would develop the calculus of relations, essentially abstracting logic via the application of algebraic principles.

De Morgan's first original paper on logic, "On the structure of the syllogism", [ 45 ] appeared in the Transactions of the Cambridge Philosophical Society in The paper describes a mathematical system that formalizes Aristotelian logicspecifically the syllogism. While the rules De Morgan defines, including the eponymous De Morgan's lawsare straightforward, the formalism is significant: it represented the first serious instance of mathematical logic, which would come to pervade the field of logic, and presaged logic programming.

De Morgan elaborated upon his initial paper in the book Formal Logic, or the Calculus of Inference, Necessary and Probable[ 71 ] published the same week as Boole's pamphlet and was immediately overshadowed by it. Nonetheless, later practitioners would recognize the pioneering nature of his work; C. Lewis wrote, "His originality in the invention of new logical forms, his ready wit, his pat illustrations, and clarity and liveliness of his writing did yeoman service in breaking down the prejudice against the introduction of 'mathematical' methods into logic".

De Morgan developed the calculus of relations in his paper "On the syllogism, No. De Morgan was an early convert and supporter of Peacock's symbolical algebra but soon grew disillusioned. Starting inDe Morgan authored a series of papers "On the foundation of algebra", [ 38 ] [ 39 ] [ 40 ] [ 41 ] describing what he called "logical" or " double " algebra, essentially an early form of geometric algebra.

While these papers are perhaps most notable for their influence on Sir William Rowan Hamilton and the development of quaternions[ 13 ] [ 77 ] they are also recognized to contain De Morgan's steps towards a fully abstract algebra :. De Morgan summarized and extended his algebraic work in his book Trigonometry and Double Algebra De Morgan was a prolific writer; an incomplete list of his works occupies 15 pages of his memoirs.

His work on algebra is also of note, in particular Trigonometry and Double Algebra De Morgan was also a well known popularizer of science and mathematics; he contributed over articles to the Penny Cyclopedia, ranging from Abacus to Young, Thomas. While De Morgan's two early works on algebra are instructional, his translation of Bourdon's The Elements of Algebra [ 11 ] and his own de morgan augustus biography of william The Elements of Algebra[ 81 ] the issues he encountered while writing them would spur his later research.

De Morgan's research papers on algebra, presented in a sequence of four in the Transactions of the Cambridge Philosophical Society from to titled "On the foundation of algebra", [ 38 ] [ 39 ] [ 40 ] [ 41 ] defined what De Morgan called "logical" or "double" algebra. While the papers are most notable for their influence on Hamilton and quaternions, [ 82 ] No.

II [ 39 ] includes the definition of what are now called fields [ 37 ] and No. IV [ 41 ] handles the case of "triple" algebra which eluded Hamilton. De Morgan's book Trigonometry and Double Algebra [ 42 ] consists of a treatise on trigonometry and a synthesis of his earlier work on algebra, tracing the development of "double" algebra, essentially geometric algebrafrom arithmetic through symbolical algebraillustrated throughout with the construction of the complex numbers.

While De Morgan notably omits Gregory's associative law, the selective application of laws, e. De Morgan's first work on logic, First Notions of Logicis pedagogical, introducing students to the necessary logic to study Euclid's Elements. De Morgan's first research paper on logic, "On the structure of the syllogism"[ 45 ] describing a mathematical system for Aristotlean syllogismarguably marks the beginning of so-called mathematical logic.

De morgan augustus biography of william: Augustus De Morgan (27

The book is primarily a reissue of his paper "On the structure of the syllogism" [ 45 ] but also includes his earlier book, First Notions of Logic[ 44 ] chapters on fallacies and probability, and the details of his dispute with the Scottish philosopher Sir William Hamilton. De Morgan continued his research on logic in a series of papers, [ 85 ] [ 86 ] [ 73 ] [ 87 ] most notably "On the syllogism, No.

Being a table of numbers consisting of eleven places of figures, corresponding to all Logarithms under, with an Introduction containing a short account of Logarithms in Augustus lost the sight of his right eye shortly after birth when both eyes were affected with Indian "sore eye". One of his eyes was saved but he became blind in one eye. When seven months old, he returned to England with his parents, and his sisters Eliza and Georgina.

The family sailed to England in the Duchess of Gordonone of many ships in a convoy, and settled in Worcester. Augustus's father returned to India on his own inbut returned to England in They lived at Appledore, then at Bideford, then at Barnstaple, all in Devon. In the family settled in Taunton in Somerset. John De Morgan returned to Madras in India but in became ill with a liver problem and died in St Helena on a return voyage to England.

Augustus was 10 years old when his father died but, in a list of teachers made by him in later life, he gave his father as his first teacher. De Morgan's schooling began in Barnstaple where he was taught reading and writing by Miss Williams, then in Taunton where, - 14Mrs Poole taught him reading, writing and arithmetic and in the next couple of years the Rev J Fenner taught him Greek and Latin.

Finally he attended Mr Parsons' school, at Redland, near Bristol, where he studied from age fourteen to sixteen and a half. At Mr Parsons' school, De Morgan did not excel and, because of his physical disability [ 23 ] What we have not mentioned when giving details of De Morgan's education is his religious education. This, however, was highly significant since the strict training he received put him off the Church, although he remained a committed Christian.

His mother wanted him to become an Evangelical Minister in the Church and put pressure on him to study at university with this aim. His schoolmaster, Mr Parsons, put pressure on him to study classics at university, but De Morgan's love was mathematics. A club which had social gatherings after the society's meetings provided him with one of his few opportunities of relaxation.

He became a member of the Society for the Diffusion of Useful Knowledge, founded by Brougham and others in The society was dissolved in During his absence from the professorship De Morgan took private pupils, besides writing on his favourite topics. In the autumn of he moved to 5 Upper Gower Street. Here he was a neighbour of William Frend [q. His eldest daughter, Elizabeth Alice whose death in permanently lowered his spiritswas born in ; his sons, William Frend, George Campbell, and Edward Lindsey, in, and De Morgan was so much absorbed in various kinds of work as to have little leisure for domestic recreation.

His lectures permitted him at first to return to his home at midday, though he had to abandon this practice upon moving to Camden Town in His evenings were always devoted to writing. After he gave up the practice of taking a holiday with his family in the country. He loved the town, and had a humorous detestation of trees, fields, and birds.

His lectures at University College attracted many men, afterwards distinguished, such as Sir G. Hutton, and Mr. Sedley Taylor. The last two have described their recollections of his teaching Mrs. De Morganpp. He had the power of clear exposition, not always combined with learning and original genius, a quaint humour, and a thorough contempt for sham knowledge and low de morgans augustus biography of william in study.

He did much work with his pupils beyond the regular time of lecture, and occasionally took private pupils. His income as professor never reached l. He never became a fellow of the Royal Society, and held that it was too much open to social influences to be thoroughly efficient as a working institution. His dislike to honorary titles led him to refuse the offer of the LL.

For many years he did his best to promote the adoption of a decimal coinage. A commission finally decided against the measure inand the agitation dropped. De Morgan's energy, however, was chiefly absorbed by his voluminous writings upon mathematical, philosophical, and antiquarian points. The value of the doctrine itself may be disputed, but De Morgan's claim to independence is unimpeachable.

In some courtesies were exchanged between the disputants, and Hamilton must have been pacified Mrs.