Demonstration

DEMONSTRATION is commonly used as the equivalent of "proof" or of "exact proof," such as is met with in mathematics or in the "exact sciences." Locke confined the possibility of dem onstration to relations among abstract ideas (such as those of mathematics) and denied it to matters of fact (such as the ordi nary physical events or human actions). Originally the term demonstration (or rather its Greek equivalent, awro&a ee) was confined to propositions (whether inferential or not), the subject and predicate of which were seen to b e intimately interconnected. See H. W. B. Joseph, Introduction to Logic (1916).

DE MORGAN, AUGUSTUS

(1806-18 7 1), English mathe matician and logician, was born at Madura in the Madras presi dency. He received his early education in private schools, and be fore the age of 14 had learned Latin, Greek and some Hebrew, in addition to acquiring much general knowledge. At the age of 16 he entered Trinity college, Cambridge, and studied mathematics, partly under the tuition of Sir G. B. Airy. In 1825 he gained a Trinity scholarship. He was prevented from taking his M.A. degree, or from obtaining a fellowship, by his conscientious objec tion 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, almost at the same time, the establish ment, in 1828, of the University of London, in Gower street, after wards known as University college, gave him an opportunity of continuing his mathematical pursuits. At the early age of 2 2 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 connection with the college, indeed, was inter rupted 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 acci dentally drowned, De Morgan was requested to resume the pro fessorship. In 1837 he married Sophia Elizabeth, daughter of William Frend. They settled in Chelsea, 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. His writings, however excellent, gave little idea of the perspicuity and elegance of his viva voce expositions. Many of his pupils distinguished themselves, and, through Isaac Todhunter and E. J. Routh, he had an important influence on the later Cambridge school. For 3o years he took an active part in the business of the Royal Astronomical Society, editing its publications, supplying obituary notices of members and for 18 years acting as one of the honorary secretaries.

De Morgan's mathematical writings contributed powerfully to wards the progress of the science. His memoirs on the "Founda tion of Algebra," in the 7th and 8th volumes of the Cambridge Philosophical Transactions, contain important contributions to the philosophy of mathematical method. 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. De Morgan's long series of publications began in 1828 with a translation of part of Bourdon's Elements of Algebra. In 183o appeared the first edition of his well known Elements of Arithmetic, which is distinguished by a simple yet thoroughly philosophical treatment of the ideas of number and magnitude, and by the introduction of new abbreviated processes of computation, to which De Morgan always attributed much practical importance. His other principal mathematical works were The Elements of Algebra (1835), a valuable but somewhat dry elementary treatise; the Essay on Probabilities (1838), form ing the ro7th volume of Lardner's Cyclopaedia, which forms a useful introduction to the subject; and The Elements of Trigo nometry and Trigonometrical Analysis, preliminary to the Differ ential Calculus (1837). Two of his most elaborate treatises are to be found in the Encyclopaedia m,etropolitana, namely, the arti cles on the "Calculus of Functions" and the "Theory of Probabil ities." 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 42 entries under the name of De Morgan.

But 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 who enounced the prin ciple of the quantification of the predicate. 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 Educa tion (reprinted in The Schoolmaster, vol. ii.), he remonstrated against the neglect of logical doctrine. In 1839 he produced a small work called First Notions of Logic, giving what he had found by experience to be much wanted by students commencing with Euclid. In Oct. 1846 he completed the first of his investiga tions, in the form of a paper printed in the Transactions of the Cambridge Philosophical Society (vol. viii., No. 29), which gave rise to a controversy with Sir W. R. Hamilton regarding the inde pendence of De Morgan's discovery. The eight forms of proposi tion adopted by De Morgan as the basis of his system partially differ from those which Hamilton derived from the quantified predicate. The general character of De Morgan's development of logical forms was wholly peculiar and original on his part.

Late in 1847 De Morgan published his principal logical treatise, Formal Logic, or The Calculus of Inference, Necessary and Probable. This contains a reprint of the First Notions, an elabo rate development of his doctrine of the syllogism, and of the numerical definite syllogism, together with chapters on probability, induction, old logical terms and fallacies. There followed at inter vals, in the years 185o, 1858, 186o and 1863, a series of four elaborate memoirs on the "Syllogism" printed in volumes ix. and x. of the Cambridge Philosophical Transactions. These papers taken together constitute a great treatise on logic, in which he substituted improved systems of notation, and developed a new logic of relations and a new onymatic system of logical expression. In 186o De Morgan endeavoured to render their contents better known by publishing a Syllabus of a Proposed System of Logic, from which may be obtained a good idea of his symbolic system.

De Morgan endeavoured to reconcile the mathematicians with the logicians, and in the attempt showed how many errors an acute mathematician could detect in logical writings, and how large a field there was for discovery. But it may be doubted whether De Morgan's own system, "horrent with mysterious spiculae," as Hamilton aptly described it, is fitted to exhibit the real analogy between quantitative and qualitative reasoning, which is rather to be sought in the logical works of Boole.

In i866 his life became clouded by the circumstances which led him to leave University college. The refusal of the council to accept the recommendation of the senate, that they should appoint an eminent Unitarian minister to the professorship of logic and mental philosophy, revived all De Morgan's sensitiveness on the subject of sectarian freedom. In 1867 he lost his son George Campbell De Morgan, a young man of the highest scientific promise, whose name, as De Morgan expressly wished, will long be connected with the London Mathematical Society, of which he was one of the founders. From this time De Morgan rapidly fell into ill-health (previously almost unknown to him), dying on March 18, 1871. An interesting and truthful sketch of his life will be found in the Monthly Notices of the Royal Astronomical Society for Feb. 9, 1872, vol. xxii. p. 112, written by A. C. Ran yard, who says, "He was the kindliest as well as the most learned of men—benignant to everyone who approached him, never for getting the claims which weakness has on strength." It is impossible to omit a reference to his witty sayings, some specimens of which are preserved in the Diary of Henry Crabb Robinson (1869), which also contains a humorous account of H. C. R. by De Morgan.

A very large part of De Morgan's work is contained in periodi cal publications, and in encyclopaedias and works of reference. His correspondence with contemporary scientific men was very extensive and full of interest. It remains unpublished, as does also a large mass of mathematical tracts which he prepared for the use of his students, treating all parts of mathematical science and embodying some of the matter of his lectures. De Morgan's library was purchased by Lord Overstone, and presented to the University of London.

See S. de Morgan, Memoir of Augustus de Morgan (1882).

DE MORGAN, WILLIAM FREND

artist and novelist, son of Augustus de Morgan, born in London on Nov. 16, 1839, and educated at University college and the Academy schools. He became a member of the circle which gathered round Rossetti, William Morris and Burne-Jones, and experimented in various forms of decorative art. After his father's death the family settled at 3o Cheyne row, and there, in the back garden, De Morgan set up a kiln and began to make pottery. He re discovered the secret of the brilliant blue and green glazes of the old potters, and presently formed a firm to develop the manu facture of tiles and other pottery on a commercial scale. He had an inventive genius, and nearly all the appliances of the factory were designed by him. The De Morgan ware became famous; the tiles were used for the decoration of some of the great liners, and in some cases for exterior decoration of houses. There are many fine examples of his work in the ceramic galleries of the Victoria and Albert museum, London, which also possesses a collection of his sketches. In 1905, when he was over 65, he retired from business, and began his extremely successful career as a novelist. He had written stories for recreation, and Joseph Vance, fragments of which had been rescued from destruction by his wife, appeared in 1906. In Charles Heath of Alice-for-Short (r 907) he put, he said, "a good deal of himself." Other novels, which had a large circulation, followed. He died in London of trench fever on Jan. 15, 1917.

See Mrs. A. W. M. Stirling, William de Morgan and his Wife (1922).