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 driven out; from this custom must be distinguished another, which consists in dismissing the souls of the dead at the close of the year and sending them on their journey to the other world; this latter custom seems to have an entirely different origin and to be due to love and not fear of the dead. In other cases it is believed that evil spirits generally or even non-personal evils such as sins are believed to be expelled. In these customs originated perhaps the scapegoat, some forms of (q.v.) and other cathartic ceremonies.

 DE MORGAN, AUGUSTUS (1806–1871), English mathematician and logician, was born in June 1806, at Madura, in the Madras presidency. His father, Colonel John De Morgan, was employed in the East India Company’s service, and his grandfather and great-grandfather had served under Warren Hastings. On the mother’s side he was descended from James Dodson, F.R.S., author of the Anti-logarithmic Canon and other mathematical works of merit, and a friend of Abraham Demoivre. Seven months after the birth of Augustus, Colonel De Morgan brought his wife, daughter and infant son to England, where he left them during a subsequent period of service in India, dying in 1816 on his way home.

Augustus De Morgan received his early education in several private schools, and before the age of fourteen years had learned Latin, Greek and some Hebrew, in addition to acquiring much general knowledge. At the age of sixteen years and a half he entered Trinity College, Cambridge, and studied mathematics, partly under the tuition of Sir G. B. Airy. In 1825 he gained a Trinity scholarship. De Morgan’s love of wide reading somewhat interfered with his success in the mathematical tripos, in which he took the fourth place in 1827. He was prevented from taking his M.A. degree, or from obtaining a fellowship, by his conscientious objection to signing the theological tests then required from masters of arts and fellows at Cambridge.

A career in his own university being closed against him, he entered Lincoln’s Inn; but had hardly done so when the establishment, in 1828, of the university of London, in Gower Street, afterwards known as University College, gave him an opportunity of continuing his mathematical pursuits. At the early age of twenty-two he gave his first lecture as professor of mathematics in the college which he served with the utmost zeal and success for a third of a century. His connexion with the college, indeed, was interrupted in 1831, when a disagreement with the governing body caused De Morgan and some other professors to resign their chairs simultaneously. When, in 1836, his successor was accidentally drowned, De Morgan was requested to resume the professorship.

In 1837 he married Sophia Elizabeth, daughter of William Frend, a Unitarian in faith, a mathematician and actuary in occupation, a notice of whose life, written by his son-in-law, will be found in the Monthly Notices of the Royal Astronomical Society (vol. v.). They settled in Chelsea (30 Cheyne Row), where in later years Mrs De Morgan had a large circle of intellectual and artistic friends.

As a teacher of mathematics De Morgan was unrivalled. He gave instruction in the form of continuous lectures delivered extempore from brief notes. The most prolonged mathematical reasoning, and the most intricate formulae, were given with almost infallible accuracy from the resources of his extraordinary memory. De Morgan’s writings, however excellent, give little idea of the perspicuity and elegance of his viva voce expositions, which never failed to fix the attention of all who were worthy of hearing him. Many of his pupils have distinguished themselves, and, through Isaac Todhunter and E. J. Routh, he had an important influence on the later Cambridge school. For thirty years he took an active part in the business of the Royal Astronomical Society, editing its publications, supplying obituary notices of members, and for eighteen years acting as one of the honorary secretaries. He was also frequently employed as consulting actuary, a business in which his mathematical powers, combined with sound judgment and business-like habits, fitted him to take the highest place.

De Morgan’s mathematical writings contributed powerfully towards the progress of the science. His memoirs on the “Foundation of Algebra,” in the 7th and 8th volumes of the Cambridge Philosophical Transactions, contain some of the most important contributions which have been made to the philosophy of mathematical method; and Sir W. Rowan Hamilton, in the preface to his Lectures on Quaternions, refers more than once to those papers as having led and encouraged him in the working out of the new system of quaternions. The work on Trigonometry and Double Algebra (1849) contains in the latter part a most luminous and philosophical view of existing and possible systems of symbolic calculus. But De Morgan’s influence on mathematical science in England can only be estimated by a review of his long series of publications, which commence, in 1828, with a translation of part of Bourdon’s Elements of Algebra, prepared for his students. In 1830 appeared the first edition of his well-known Elements of Arithmetic, which did much to raise the character of elementary training. It is distinguished by a simple yet thoroughly philosophical treatment of the ideas of number and magnitude, as well as by the introduction of new abbreviated processes of computation, to which De Morgan always attributed much practical importance. Second and third editions were called for in 1832 and 1835; a sixth edition was issued in 1876. De Morgan’s other principal mathematical works were The Elements of Algebra (1835), a valuable but somewhat dry elementary treatise; the Essay on Probabilities (1838), forming the 107th volume of Lardner’s Cyclopaedia, which forms a valuable introduction to the subject; and The Elements of Trigonometry and Trigonometrical Analysis, preliminary to the Differential Calculus (1837). Several of his mathematical works were published by the Society for the Diffusion of Useful Knowledge, of which De Morgan was at one time an active member. Among these may be mentioned the Treatise on the Differential and Integral Calculus (1842); the Elementary Illustrations of the Differential and Integral Calculus, first published in 1832, but often bound up with the larger treatise; the essay, On the Study and Difficulties of Mathematics (1831); and a brief treatise on Spherical Trigonometry (1834). By some accident the work on probability in the same series, written by Sir J. W. Lubbock and J. Drinkwater-Bethune, was attributed to De Morgan, an error which seriously annoyed his nice sense of bibliographical accuracy. For fifteen years he did all in his power to correct the mistake, and finally wrote to The Times to disclaim the authorship. (See Monthly Notices of the Royal Astronomical Society, vol. xxvi. p. 118.) Two of his most elaborate treatises are to be found in the Encyclopaedia metropolitana, namely the articles on the Calculus of Functions, and the Theory of Probabilities. De Morgan’s minor mathematical writings were scattered over various periodicals. A list of these and other papers will be found in the Royal Society’s Catalogue, which contains forty-two entries under the name of De Morgan.

In spite, however, of the excellence and extent of his mathematical writings, it is probably as a logical reformer that De Morgan will be best remembered. In this respect he stands alongside of his great contemporaries Sir W. R. Hamilton and George Boole, as one of several independent discoverers of the all-important principle of the quantification of the predicate. Unlike most mathematicians, De Morgan always laid much stress upon the importance of logical training. In his admirable papers upon the modes of teaching arithmetic and geometry, originally published in the Quarterly Journal of Education (reprinted in The Schoolmaster, vol ii.), he remonstrated against the neglect of