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Augustus De Morgan
De Morgan Augustus.jpg
Born (1806-06-27)27 June 1806
Madurai, Carnatic, Madras Presidency, (present-day India)
Died 18 March 1871(1871-03-18) (aged 64)
London, England
Nationality British
Alma mater Trinity College, Cambridge
Known for De Morgan's laws
De Morgan algebra
De Morgan hierarchy
Relation algebra
Universal algebra
Scientific career
Fields Mathematician and logician
Institutions University College London
University College School
Academic advisors John Philips Higman
George Peacock
William Whewell
Notable students Edward Routh
James Joseph Sylvester
Frederick Guthrie
William Stanley Jevons
Ada Lovelace
Francis Guthrie
Stephen Joseph Perry
Influences George Boole
Influenced Thomas Corwin Mendenhall
Isaac Todhunter
He was the father of William De Morgan.

Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous.



Augustus De Morgan was born in Madurai, in the Carnatic region of India in 1806. His father was Lieut.-Colonel John De Morgan (1772–1816), who held various appointments in the service of the East India Company, and his mother, Elizabeth (née Dodson, 1776–1856), was daughter of John Dodson and granddaughter of James Dodson, who computed a table of anti-logarithms (inverse logarithms). Augustus De Morgan became blind in one eye a month or two after he was born. The family moved to England when Augustus was seven months old. As his father and grandfather had both been born in India, De Morgan used to say that he was neither English, nor Scottish, nor Irish, but a Briton "unattached", using the technical term applied to an undergraduate of Oxford or Cambridge who is not a member of any one of the Colleges.

When De Morgan was ten years old his father died. Mrs De Morgan resided at various places in the southwest of England, and her son received his primary education at various schools of no great account. His mathematical talents went unnoticed until he was fourteen, when a family-friend discovered him making an elaborate drawing of a figure from one of Euclid's works with a ruler and compasses.

He received his secondary education from Mr Parsons, a fellow of Oriel College, Oxford, who appreciated classics better than mathematics. His mother was an active and ardent member of the Church of England, and desired that her son should become a clergyman, but by this time De Morgan had begun to show his non-conforming disposition. He became an atheist.

University education

In 1823, at the age of sixteen, he entered Trinity College, Cambridge, where he came under the influence of George Peacock and William Whewell, who became his lifelong friends; from the former he derived an interest in the renovation of algebra, and from the latter an interest in the renovation of logic—the two subjects of his future life work. His college tutor was John Philips Higman, FRS (1793–1855).

At college he played the flute for recreation and was prominent in the musical clubs. His love of knowledge for its own sake interfered with training for the great mathematical race; as a consequence he came out fourth wrangler. This entitled him to the degree of Bachelor of Arts; but to take the higher degree of Master of Arts and thereby become eligible for a fellowship it was then necessary to pass a theological test. To the signing of any such test De Morgan felt a strong objection, although he had been brought up in the Church of England. In about 1875 theological tests for academic degrees were abolished in the Universities of Oxford and Cambridge.

London University

As no career was open to him at his own university, he decided to go to the Bar, and took up residence in London; but he much preferred teaching mathematics to reading law. About this time the movement for founding London University (now University College London) took shape. The two ancient universities of Oxford and Cambridge were so guarded by theological tests that no Jew or Dissenter outside the Church of England could enter as a student, still less be appointed to any office. A body of liberal-minded men resolved to meet the difficulty by establishing in London a university on the principle of religious neutrality. De Morgan, then 22 years of age, was appointed professor of mathematics. His introductory lecture "On the study of mathematics" is a discourse upon mental education of permanent value, and has been recently reprinted in the United States.

The London University was a new institution, and the relations of the Council of management, the Senate of professors and the body of students were not well defined. A dispute arose between the professor of anatomy and his students, and in consequence of the action taken by the council, several professors resigned, headed by De Morgan. Another professor of mathematics was appointed, who then died a few years later. De Morgan had shown himself a prince of teachers: he was invited to return to his chair, which thereafter became the continuous centre of his labours for thirty years.

The same body of reformers—headed by Lord Brougham, a Scotsman eminent both in science and politics who had instituted the London University—founded about the same time a Society for the Diffusion of Useful Knowledge. Its object was to spread scientific and other knowledge by means of cheap and clearly written treatises by the best writers of the time. One of its most voluminous and effective writers was De Morgan. He wrote a great work on The Differential and Integral Calculus which was published by the Society; and he wrote one-sixth of the articles in the Penny Cyclopedia, published by the Society, and issued in penny numbers. When De Morgan came to reside in London he found a congenial friend in William Frend. Both were arithmeticians and actuaries, and their religious views were somewhat similar. Frend lived in what was then a suburb of London, in a country-house formerly occupied by Daniel Defoe and Isaac Watts. De Morgan with his flute was a welcome visitor.

Jonardon Ganeri has observed that it was this period of the mid-nineteenth century pointed to by Mary Boole that saw George Boole (1815–1864) and Augustus De Morgan (1806–1871) make their pioneering applications of algebraic ideas to the formulation of logic (algebraic logic and Boolean logic), and has suggested that these figures were likely to have been aware of the Indian system of logic, and in turn, that their awareness of the shortcomings of propositional logic as it was then formulated may have contributed to their willingness to look beyond their own logical tradition.


Augustus was one of seven children, four of whom survived to adulthood.

  • Eliza (1801–1836) married Lewis Hensley, a surgeon, living in Bath.
  • Augustus (1806–1871)
  • George (1808–1890), a barrister-at-law who married Josephine, daughter of Vice Admiral Josiah Coghill, 3rd Baronet Coghill
  • Campbell Greig (1811–1876), a surgeon at the Middlesex Hospital

In the autumn of 1837, he married Sophia Elizabeth Frend (1809–1892), eldest daughter of William Frend (1757–1841) and Sarah Blackburne (1779–?), a granddaughter of Francis Blackburne (1705–1787), Archdeacon of Cleveland.

De Morgan had three sons and four daughters, including fairytale author Mary De Morgan. His eldest son was the potter William De Morgan. His second son George acquired distinction in mathematics at University College and the University of London. He and another like-minded alumnus conceived the idea of founding a mathematical society in London, where mathematical papers would be not only received (as by the Royal Society) but actually read and discussed. The first meeting was held in University College; De Morgan was the first president, his son the first secretary. It was the beginning of the London Mathematical Society.

Retirement and death

Augustus De Morgan
Augustus De Morgan.

In 1866 the chair of mental philosophy in University College fell vacant. James Martineau, a 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. 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. His pupils secured him a pension of £500 p.a., but misfortunes followed. Two years later his son George—the "younger Bernoulli", as Augustus loved to hear him called, 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 1871.

Mathematical work

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, a Scotsman, professor of logic and metaphysics at the University of Edinburgh; the other was a knight (that is, won the title), an 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 Hamilton, contributed to mathematics, especially geometric algebra, and first described the Quaternions. De Morgan was interested in the work of both, and corresponded with both; the correspondence with the Irishman was marked by friendship and terminated only by death.

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 never sought to become a Fellow of the Royal Society, and he never attended a meeting of the Society. He never voted at an election, and he never visited the House of Commons, the Tower of London or Westminster Abbey.

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 Algebra, published in 1849; and his earlier view of formal logic is found in a volume published in 1847. His most distinctive work is styled A Budget of Paradoxes; it originally appeared as letters in the columns of the Athenæum journal; it was revised and extended by De Morgan in the last years of his life, and was published posthumously by his widow.

George Peacock's theory of algebra was much improved by D. F. Gregory, a 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.


De Morgan later in his life became interested in the phenomena of spiritualism. In 1849, he had investigated clairvoyance and was impressed by the subject. He later carried out paranormal investigations in his own home with the American medium Maria Hayden. The result of those investigations was later published by his wife Sophia. De Morgan believed that his career as a scientist might have been affected if he had revealed his interest in the study of spiritualism, so he helped to publish the book anonymously. The book was published in 1863, titled From Matter to Spirit: The Result of Ten Years Experience in Spirit Manifestations.

According to historian Janet Oppenheim, De Morgan's wife Sophia was a convinced spiritualist but De Morgan shared a third way position on spiritualist phenomena, which Oppenheim defined as a "wait-and-see position"; he was neither a believer nor a sceptic. Instead, his viewpoint was that the methodology of the physical sciences does not automatically exclude psychic phenomena, and that such phenomena may be explainable in time by the possible existence of natural forces which physicists had not yet identified.

Psychical researcher John Beloff wrote that De Morgan was the first notable scientist in Britain to take an interest in the study of spiritualism and his studies had influenced the decision of William Crookes to also study spiritualism. Beloff also claims that De Morgan was an atheist and so he was debarred from a position at Oxford or Cambridge.


Beyond his great mathematical legacy, the headquarters of the London Mathematical Society is called De Morgan House and the student society of the Mathematics Department of University College London is called the Augustus De Morgan Society.

The crater De Morgan on the Moon is named after him.

Selected writings

  • De Morgan, Augustus (1836). An Explanation of the Gnomonic Projection of the Sphere. London: Baldwin.
  • De Morgan, Augustus (1837a). Elements of Trigonometry, and Trigonometrical Analysis. London: Taylor & Walton.
  • De Morgan, Augustus (1837b). The Elements of Algebra. London: Taylor & Walton.
  • De Morgan, Augustus (1838). An Essay on Probabilities, and Their Application to Life Contingencies and Insurance Offices. London: Longman, Orme, Brown, Green & Longmans.
  • De Morgan, Augustus (1840a). The Elements of Arithmetic. London: Taylor & Walton.
  • De Morgan, Augustus (1840b). First Notions of Logic, Preparatory to the Study of Geometry. London: Taylor & Walton.
  • De Morgan, Augustus (1842). The Differential and Integral Calculus. London: Baldwin.
  • De Morgan, Augustus (1845). The Globes, Celestial and Terrestrial. London: Malby & Co..
  • De Morgan, Augustus (1847). Formal Logic or The Calculus of Inference, Necessary and Probable. London: Taylor & Walton..
  • De Morgan, Augustus (1849). Trigonometry and Double Algebra. London: Taylor, Walton & Malbery.
  • De Morgan, Augustus (1860). Syllabus of a Proposed System of Logic. London: Walton & Malbery..
  • De Morgan, Augustus (1872). A Budget of Paradoxes. London: Longmans, Green.

See also

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