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Rh 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 Airy, subsequently the astronomer royal. In 1825 he gained a Trinity scholarship. De Morgan's attention was by no means confined to mathematics, and his love of wide reading somewhat interfered with his success in the mathematical tripos, in which he took the fourth place in 1827, before he had completed his twenty-first year. He was prevented from taking his M. A. degree, or from obtaining a fellowship, to which he would doubtless have been elected, by his conscientious objection to signing the theological tests then required from masters of arts and fellows at Cambridge. A strong repugnance to any sectarian restraints upon the freedom of opinion was one of De Morgan's most marked characteristics throughout life.

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 years he gave his first lecture as professor of mathematics in a college which he served with the utmost zeal and success for a third of a century. His connection 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 Mr White was accidentally drowned, De Morgan was requested to resume the professorship. It may be added that his choice of a literary and scientific career was made against the advice of his relatives and friends, who, on his entering Lincoln's Inn, confidently anticipated for him a distinguished and lucrative career at the bar.

In 1837 De Morgan 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). Henceforward De Morgan's life is scarcely more than a record of his constant labours, and his innumerable publications. As in the case of many scholars, the even tenor of his life was unbroken by remarkable incidents. Surrounded by a growing family, ultimately seven in number, he sought happiness in his home, in his library, and in the energetic and vigorous discharge of his college duties. He seldom travelled or enjoyed relaxation, and could with difficulty be induced to remain many days from home.

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 formulæ, 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 Mr Todhunter and Mr Routh, he has had an important influence on the modern Cambridge school. In addition to occasional extra courses, it was his habit to give two lectures on each of the six week days throughout the working session of thirty weeks or more. Each lecture was exactly one hour and a quarter in length, and at the close a number of questions and problems were always given, to which the pupils returned written answers. These were all corrected by the professor's own hand, and personal explanations given before or after the lecture.

Although the best hours of the day were thus given to arduous college work, his public labours in other directions were extensive. 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 18 years acting as one of the honorary secretaries. His work for this society alone, it is said, would have been occupation enough for an ordinary man. 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, published by De Morgan in 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 has been widely used in schools, and has done 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, and more than 20,000 copies have been sold; the book is still in use, a sixth edition having been 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, still much used, being probably the best simple introduction to the theory in the English language; 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 great Treatise on the Differential and Integral Calculus, 1842, which still remains the most extensive and complete English treatise on the subject; the Elementary Illustrations of the Differential and Integral Calculus, first published in 1832, but often bound up with the larger treatise; the valuable 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 Lubbock and 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 Encyclopædia Metropolitana, namely the articles on the