Morgan, Augustus De

(1806-1871), mathematician and historian

by Leslie Stephen, rev. I. Grattan-Guinness

© Oxford University Press 2004 All rights reserved

Morgan, Augustus De (1806-1871), mathematician and historian, fifth child of Colonel John De Morgan (1772-1816) of the Indian army, was born on 27 June 1806 at Madura in the far south of India. His mother, Elizabeth (1776-1856), was the daughter of John Dodson of the custom house, and granddaughter of James Dodson. Seven months after De Morgan's birth his family sailed for England with their five children, settling at Worcester. The youngest son, Campbell Greig De Morgan, born in 1811, made his career in medicine.

De Morgan was educated privately and sent to various schools, one of his teachers being J. Fenner, a Unitarian minister and an uncle of H. Crabb Robinson. He pricked out equations on a pew at St Michael's Church, Bristol, some of which remained after his death, instead of listening to the sermon. In February 1823 he entered Trinity College, Cambridge, where he soon showed his mathematical ability. He made many friends at college, including his teachers William Whewell and George Peacock. He belonged to a musical society called the Camus (Cambridge Amateur Musical Union Society), and was a skilful flautist. In 1827 he graduated as fourth wrangler, though far superior in mathematical ability to any man in his year. He was disappointed by the result, which was due to his discursive reading. He always disliked competitive examinations as tending to give the advantage to the docile over the original students, and because they encouraged 'cramming'.

De Morgan's career was now affected by religious issues. His parents had inculcated their strict religious principles into him from a very early age; but even at school he had revolted from the doctrines held by his mother, and at Cambridge he became heterodox. He was throughout life a strong theist and preferred the Unitarian to other creeds, but he never definitely joined any church, calling himself a 'Christian unattached'. His scruples prevented him from proceeding to the MA degree or becoming a candidate for a fellowship. After some thoughts of medicine he resolved to go to the bar, and entered Lincoln's Inn.

Professor of mathematics
At this time London University was established, without adherence to any religious institution. De Morgan found law unpalatable, and, though the youngest applicant, he was unanimously elected on 23 February 1828 the first professor of mathematics on the strength of very high testimonials from Peacock, G. B. Airy, and other Cambridge authorities. He gave his introductory lecture, 'On the study of mathematics', on 5 November 1828. However, difficulties soon arose in the working of the new institution: the council claimed the right of dismissing a professor without assigning reasons, and so dismissed the professor of anatomy. De Morgan immediately resigned his post in a letter dated 24 July 1831.

In October 1836 his successor, G. J. P. White, was accidentally drowned, whereupon De Morgan offered himself as a temporary substitute. He was then invited to resume the chair and, as the regulations had by then been so altered as to give the necessary independence to professors, he accepted the invitation. He was reappointed and served as professor for the next thirty years. At that time King's College, London, was being formed as an institution under the Church of England, London University became University College, and the two colleges merged as the University of London, an examining body which conferred degrees.

De Morgan revealed a power of clear exposition, often though not always combined with learning and original genius, a quaint sense of humour, and a thorough contempt for sham knowledge and low aims in study. He did much work with his students beyond regular lecture times, and occasionally took private pupils. His income as professor never reached £500, and in later years declined, seldom exceeding £300.

In the autumn of 1831 De Morgan moved to 5 Upper Gower Street, where he was a neighbour of William Frend, mathematician and political reformer. He married Frend's daughter, Sophia Elizabeth (1809-1892) on 3 August 1837, and settled in 69 Gower Street. Of his seven children, William Frend De Morgan (1839-1917), was to make a distinguished career as a ceramicist and novelist, while George Campbell De Morgan [see below] showed great promise in mathematics before his early death. Augustus De Morgan was so much absorbed in various kinds of work that he had little leisure for domestic recreation. His lectures permitted him at first to return home at midday, but this practice had to be abandoned on his move to 7 Camden Street, Camden Town, in 1844. After 1840 he gave up the practice of taking a holiday with his family in the country. An inveterate Londoner, he loved the town, and had a humorous detestation of trees, fields, and birds. He could not even bear Blackheath, calling the heath 'desolation', although he liked the steamboats.

Membership of societies
From the first De Morgan was a most energetic worker in other London institutions. Elected in May 1828 a fellow of the Astronomical Society, he was placed on its council in 1830. He was secretary from 1831 to 1838 and from 1848 to 1854, and at other periods held office as vice-president. He took a keen interest in its proceedings, edited its publications, and made many intimate friends at its meetings. By contrast, 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. He did, however, become a member of the Society for the Diffusion of Useful Knowledge (SDUK), founded by Lord Brougham and others in 1826. It published some of his early writings, and he contributed a great number of articles to its other publications: the Penny Cyclopaedia, for which he wrote about 850 of the articles, the Quarterly Journal of Education, and the unfortunately short-lived Biographical Dictionary. He also published with the society an Elements of Algebra (1835), and, especially, a massive and high-level Treatise on the Differential and Integral Calculus, which appeared in instalments between 1836 and 1842. He became a member of the committee in 1843. (The society was dissolved in 1846.)

For many years De Morgan promoted the adoption of a decimal coinage. He gave evidence before commissions, and was on the council of the Decimal Association formed in 1854. A commission finally decided against the measure in 1859 and the agitation dropped.

Development of logic
De Morgan read and wrote much despite losing an eye in childhood. In a brilliant remark made in a book review in 1868 he commented that mathematics and logic constituted the 'two eyes of exact science', but that each was trying to put out the other one, 'believing that it can see better with one eye than with two' (The Athenaeum, 2, 1868, 71-3). This point surrounded much of his research work, for his mathematical speciality was algebra, and his contributions to logic drew upon certain algebraic principles.

As an undergraduate De Morgan fell under the influence of Peacock and Whewell; from them he studied the distinction, then widely discussed in Britain, between form and matter in common algebra. Form related to the general conception of an algebra, like his 'double algebra' where the elements could be combined in two different ways (like + and × in ordinary arithmetic); his work here helped W. R. Hamilton to develop his quaternion algebra in 1843. Matter arose in particular cases such as the 'real' existence of negative numbers.

In 1836, De Morgan wrote for the Encyclopaedia metropolitana the first comprehensive survey of functional equations, a fairly new algebra which had been brought into English mathematics in the 1810s by John Herschel and especially Charles Babbage. The task was to find mathematical functions which obeyed some given particular property. His work was to relate to the last of his four main contributions to logic, which he had begun to study at this time and most of which appeared in a suite of five papers, 'On the syllogism', published between 1846 and 1862, and in his book Formal Logic (1847).

First, De Morgan published in 1831 a book, On the Study and Difficulties of Mathematics, followed in 1839 by a booklet, First Notions of Logic (Preparatory to the Study of Geometry), in which he tried to discern the logical manoeuvres involved in Euclid's logic. The status of the parallel postulate in geometry concerns the axiomatization of the theory, which is a different question.

De Morgan was then, in the 1840s, involved in a priority dispute initiated by the Scottish philosopher Sir William Hamilton over the 'quantification of the predicate', in which propositions such as 'all Xs are some Ys' were added to syllogistic logic. They greatly expanded the range of valid and invalid modes. A pair of laws now named after him arose in this context in 1853, in his third paper: 'The contrary of an aggregate [the complement of the union of two classes] is the compound [intersection] of the contraries of the [two] aggregants; the contrary of a compound is the aggregate of the contraries of the components'. However, he did not realize their full importance for logic.

De Morgan's third contribution was his examination of various aspects of his expanded syllogistic logic. These included 'numerically definite' ones such as 'Each of the 50 Xs is one or other of the 70 Ys'. He also studied algebra-like properties of the copula such as transitivity, and stressed its various roles, such as predication distinct from identity. He noted analogies with algebra, such as logical inference with algebraic elimination. He devised ingenious semiotic notations using dots and brackets to represent propositions--for example, for predicates X and Y: 'X).(Y' for 'No X is Y', and 'X(.)Y' for 'Y sometimes not in X'. (His handling of classes of objects satisfying predicates was of the part-whole kind, then the usual way of treating collections of things.) These notations also provided concise ways of representing valid modes.

De Morgan's fourth paper, On the Logic of Relations, published in 1860, contained arguably his most substantial innovation in logic. He outlined many of the basic properties of a logic of two-place relations, such as 'brother of' or 'greater than', and operations such as composition and inversion (if possible) upon them. He deployed algebraic notations to signify many of his notions and findings. From a structural point of view this work showed close kinship with his study of functional equations; several principal properties, for example, the existence of inverse(s), are common to both.

Most of De Morgan's work in logic can be characterized as conservative (though valuable) extensions of syllogistic logic--less radical than the efforts of his friend George Boole in 1847. But he was well aware of the limitations of syllogism, not only the absence of relations but also non-syllogistic inferences such as 'man is animal, therefore the head of a man is the head of an animal'.

On the more informal side of logic, De Morgan produced A Budget of Paradoxes, which was based on many short articles in The Athenaeum. The book was published in 1872 shortly after his death, under the editorship of his widow. He assembled many newspaper reports, short articles, poems, and much other material, commenting with logical acumen and wit.

In 1837 De Morgan wrote another article for the Encyclopaedia metropolitana, on probability theory. He considered probabilistic inference, that is, the transmission of probability values from premises to consequences. In An Essay on Probabilities of the following year he emphasized the importance of the subject in life contingencies, which gave his book a good reception in the assurance industry. Indeed, during 1831-6 when he was absent from the university, he often worked as an actuary; he also contributed to the Insurance Record.

Other writing
De Morgan also wrote extensively on the history of mathematics and astronomy, including a long series of articles in the journal Companion to the Almanac and many of the SDUK pieces in the Quarterly Journal of Education and the Penny Cyclopaedia. His two most substantial topics were an annotated bibliography of books in arithmetic, information which he gathered together in Arithmetical Books (1847), and articles in which he exposed for the first time the dishonest way in which Isaac Newton had argued within the Royal Society in the early eighteenth century that G. W. Leibniz had plagiarized him in the invention of the calculus. Until the First World War, De Morgan's historical work was well remembered; in particular, it was cited in many articles in the Dictionary of National Biography. He was a prolific correspondent, often adorning his letters with well-drawn caricatures and sketches. He had a love of puns, and of ingenious puzzles and paradoxes. A year before his death an annuity of £100 was obtained from the government, and accepted with some reluctance.

Among other activities De Morgan collaborated with his wife on pioneering studies of psychical mediumship. As 'C. D.' she published From Matter to Spirit in 1863, to which her husband, 'A. B.', contributed an excellent preface.

In 1866 the chair of mental philosophy and logic at University College became vacant. A discussion arose as to the principle of religious neutrality avowedly adopted by the college when refusing to appoint as professor of philosophy James Martineau, who, as a Unitarian minister, was pledged to maintain the creed of a particular sect. De Morgan, on the other hand, held that any consideration of a candidate's ecclesiastical position or religious creed was inconsistent with the principle, so again resigned his office, in a letter dated 10 November 1866. Though no personal bitterness was produced, De Morgan had felt the blow so keenly that it injured his health. Some of his old students begged him to allow his portrait to be painted for the library of 'our old college'. But he objected on principle to testimonials, and is alleged to have replied that 'our old college no longer exists'.

De Morgan died on 18 March 1871 at his home, 6 Merton Road, Hampstead, and was buried in Kensal Green cemetery. His library consisted at the end of his life of about three thousand volumes. He was a genuine book-hunter, though his means compelled him to limit himself to occasional treasures from bookstalls. He made many amusing marginal and learned annotations, and turned his bibliographical researches to a good account in his historical writings. His library was bought after his death by Lord Overstone and presented to the University of London, where it became part of the University Library at its foundation.

De Morgan's son George Campbell De Morgan (1841-1867) was born on 16 August 1841 at Gower Street. He was educated at University College School, 1856-7, then under his father at University College, 1857-60, graduating MA in 1863 with the award of a gold medal. He was appointed mathematics master at the school in 1866; at the time of his death he was vice-principal of University Hall. As well as producing some promising mathematical work, he was one of the founders in 1865 of the London Mathematical Society (and his father was the first president). He died on 14 October 1867 at Ventnor, Isle of Wight.


S. E. De Morgan, Memoirs of Augustus De Morgan (1882)
M. Panteki, 'Relationships between algebra, logic and differential equations in England, 1800-1860', PhD diss., Middlesex University (CNAA), 1992
A. Rice, 'Augustus De Morgan and the development of university mathematics in London in the nineteenth century', PhD diss., Middlesex University, 1997
A. Rice, 'Inspiration or desperation? Augustus De Morgan's appointment to the chair of mathematics at London University in 1828', British Journal for the History of Science, 30 (1997), 257-74
A. Rice, 'Mathematics in the metropolis: a survey of Victorian London', Historia Mathematica, 23 (1996), 376-417
A. Rice, R. J. Wilson, and J. H. Gardner, 'From student club to national society: the founding of the London Mathematical Society in 1865', Historia Mathematica, 22 (1995), 402-11
A. Rice, 'Augustus De Morgan: historian', History of Science, 34 (1996), 201-40
P. Heath, ed., On the syllogism, and other logical writings (1966)
A. De Morgan, A budget of paradoxes (1872)
d. cert.
b. cert. [G. C. De Morgan]
d. cert. [G. C. De Morgan]

American Philosophical Society, Philadelphia, letters
Bodl. Oxf., corresp.; family corresp.
LUL, corresp. and papers
RAS, corresp. and papers; letters to RAS
Royal Observatory Library, Edinburgh, catalogue of mathematical books
UCL, corresp. and papers
UCL, letters to Society for the Diffusion of Useful Knowledge |  BL, corresp. with Charles Babbage, Add. MSS 37185-37200, passim
Bodl. Oxf., corresp. with Lady Byron
CUL, corresp. with Sir George Airy
CUL, letters to William Hepworth Dixon
CUL, letters to Lord Kelvin
NL Scot., letters to Alexander Campbell Fraser
RS, corresp. with Sir John Herschel
RS, letters to Sir John Lubbock
TCD, letters to Sir William Hamilton
Trinity Cam., letters to William Whewell
U. Edin. L., special collections division, letters to James Halliwell-Phillips
UCL, corresp. with George Boole
UCL, corresp. with Lord Brougham
UCL, letters to F. Hendriks

T. Woolner, marble bust, after 1876, U. Lond.
A. De Morgan, self-portrait, pen-and-ink drawing, RAS
Maull & Polyblank, carte-de-visite, NPG
J. J. E. Mayall, photograph, NPG [see illus.]
carte-de-visite, NPG
wood-engraving, NPG; repro. in ILN (22 April 1854)

Wealth at death  
under £7000: probate, 29 April 1871, CGPLA Eng. & Wales

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