Calculus of Functions, and the Theory of Probabilities. The former article contains a profound investigation into the principles of symbolic reasoning; the latter is still the most complete mathematical treatise on the subject in the English language, giving as it does a resumé of Laplace's Théorie Analytique des Probabilités. De Morgan's minor mathematical writings are scattered over various periodicals; five papers will be found in the Cambridge Mathematical Journal, ten in the Cambridge and Dublin Mathematical Journal, several in the Philosophical Magazine, while others of more importance are printed in the Cambridge Philosophical Transactions. 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.
In spite of the excellence and extent of his mathematical writings, it is probably as a logical reformer that De Morgan will be best known to future times. In this respect he stands alongside of his great contemporaries Hamilton and 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 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 October 1846 he completed the first of his original investigations, in the form of a paper printed in the Transactions of the Cambridge Philosophical Society (vol. viii. No. 29). In this paper the principle of the quantified predicate was referred to, and there immediately ensued a memorable controversy with Sir W. Hamilton regarding the independence of De Morgan's discovery, some communications having passed between them in the autumn of 1846. The details of this dispute will be found by those interested in the original pamphlets, in the Athenæum newspaper, or in the appendix to De Morgan's Formal Logic. Suffice it to say that the independence of De Morgan's discovery was subsequently recognized by Hamilton, and that those acquainted with De Morgan's character could never suppose that it was otherwise. Moreover, the eight forms of proposition 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.
Not a year passed before De Morgan, late in 1847, published his principal logical treatise, called Formal Logic, or the Calculus of Inference, Necessary and Probable. This contains a reprint of the First Notions, an elaborate development of his doctrine of the syllogism, and of the numerically definite syllogism, together with chapters of great interest on probability, induction, old logical terms, and fallacies. The severity of the treatise is relieved by characteristic touches of humour, and by quaint anecdotes and allusions furnished from his wide reading and perfect memory.
There followed at intervals, in the years 1850, 1858, 1860, 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. Apart, however, from their principal purpose, these memoirs are replete with acute remarks, happy illustrations, and abundant proofs of De Morgan's varied learning. Unfortunately these memoirs are accessible to few readers, otherwise they would form invaluable reading for the logical student. In 1860 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, but the more readable and interesting discussions contained in the memoirs are of necessity omitted. The article "Logic" in the English Cyclopædia (1860) completes the list of his logical publications.
Throughout his logical writings De Morgan was led by the idea that the followers of the two great branches of exact science, logic and mathematics, had made blunders,—the logicians in neglecting mathematics, and the mathematicians in neglecting logic. He endeavoured to reconcile them, 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 spiculæ," 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. (See Boole, vol. iv. p. 47.)
Perhaps the largest part, in volume, of De Morgan's writings remains still to be briefly mentioned; it consists of detached articles contributed to various periodical or composite works. During the years 1833-43, he contributed very largely to the first edition of the Penny Cyclopædia, writing chiefly on mathematics, astronomy, physics, and biography. His articles of various length cannot be less in number than 850, as may be ascertained from a signed copy in the British Museum, and they have been estimated to constitute a sixth part of the whole Cyclopædia, of which they formed perhaps the most valuable portion. He also wrote biographies of Newton and Halley for Knight's British Worthies, various notices of scientific men for the Gallery of Portraits, and for the uncompleted Biographical Dictionary of the Useful Knowledge Society, and at least seven articles in Smith's Dictionary of Greek and Roman Biography.
Some of De Morgan's most interesting and useful minor writings are to be found in the Companions to the British Almanack, to which he contributed without fail one article each year from 1831 up to 1857 inclusive. In these carefully written papers he treats a great variety of topics relating to astronomy, chronology, decimal coinage, life-assurance, bibliography, and the history of science. Most of them are as valuable now as when written.
Among De Morgan's miscellaneous writings may be mentioned his Explanation of the Gnomonic Projection of the Sphere, 1836, including a description of the maps of the stars, published by the Useful Knowledge Society; his Treatise on the Globes, Celestial and Terrestrial, 1845; and his remarkable Book of Almanacks, (second edition 1871), which contains a series of 35 almanacks, so arranged with indices of reference, that the almanack for any year, whether in old style or new, from any epoch, ancient or modern, up to 2000 A.D., may be found without difficulty, means being added for verifying the almanack and also for discovering the days of new and full moon from 2000 B.C. up to 2000 A.D. De Morgan expressly draws attention to the fact that the plan of this book was that of Francœur and Ferguson, but the plan was developed by one who was an unrivalled master of all the intricacies of chronology. The two best tables of logarithms, the small five-figure tables of the Useful Knowledge Society (1839 and 1857), and Shroen's Seven Figure-Table (5th ed. 1865), were printed under De Morgan's superintendence. Several works edited by him will be found mentioned in the British Museum Catalogue. His numerous anonymous contributions through a long series of years to the Athenæum, and to Notes and Queries, and his occasional articles in the North British Review, Macmillan's Magazine, &c., must be passed over with this bare mention.
Considerable labour was spent by De Morgan upon the subject of decimal money. He was a great advocate of the pound and mil scheme. His evidence on this subject was sought by the Royal Commission, and, besides constantly supporting the Decimal Association in periodical publications, he published several separate pamphlets on the subject.
One marked character of De Morgan was his intense and yet reasonable love of books. He was a true bibliophil, and loved to surround himself, as far as his means allowed, with curious and rare books. He revelled in all the mysteries of watermarks, title pages, colophons, catch-words, and the like; yet he treated biblio-