Augustus de Morgan and George Boole s Contribution to Digital Electronics Essay Example for Free

Augustus de Morgan and George Boole s Contribution to Digital Electronics Essay Augustus De Morgan 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 Companys service, and his grand father and great-grandfather had served under Warren Hastings. On the mothers side he was descended from JamesDodson,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 Morgans 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. 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 Morgans 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. I. n 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 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 investigations, in the form of a paper printed in the Transactions of the Cambridge Philosophical Society (vol. iii. No. 29). In this paper the principle of the quantified predicate was referred to, and there immediately ensued a memorable controversy with Sir W. R. Hamilton regarding the independence of De Morgans discovery, some communications having passed between them in the autumn of 1846. The details of this dispute will be found in the original pamphlets, in the [[Athenaeum]] and in the appendix to De Morgans Formal Logic. Suffice it to say that the independence of De Morgans discovery was subsequently recognized by Hamilton. 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 Morgans development of logical forms was wholly peculiar and original on his part. Late in 1847 De Morgan 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 numerical 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. nd 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 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 Cyclopaedia (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 Morgans 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. Perhaps the largest part, in volume, of De Morgans writings remains still to be briefly mentioned; it consists of detached articles contributed to various periodical or composite works. During the years 1833-1843 he contributed very largely to the first edition of the [[Penny]] Cyclopaedia, writing chiefly on mathematics, astronomy, physics and biography. His articles of various length cannot be less in number than 850, and they have been estimated to constitute a sixth part of the whole Cyclopaedia, of which they formed perhaps the most valuable portion. He also wrote biographies of Sir Isaac Newton and Edmund Halley for Knights 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 Smiths Dictionary of Greek and Roman Biography. Some of De Morgans 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 Morgans 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 Know ledge Society; his Treatise on the Globes, Celestial and Terrestrial,1845, and his remarkable [[Book]] of Almanacks (2nd edition, 1871), which contains a series of thirty-five almanacs, so arranged with indices of reference, that the almanac for any year, whether in old style or new, from any epoch, ancient or modern, up to A.  D. 2000, may be found without difficulty, means being added for verifying the almanac and also for discovering the days of new and full moon from 2000 B. c. up to A. D. 2000. De Morgan expressly draws attention to the fact that the plan of this book was that of L. B. Francoeur and J. 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 Shroens Seven Figure-Table (5th ed. , 1865), were printed under De Morgans superintendence. Several works edited by him will be found mentioned in the British Museum Catalogue. He made numerous anonymous contributions through a long series of years to the Athenaeum, and to Notes and Queries, and occasionally to The North British Review, Macmillans Magazine, c.  Considerable labour was spent by De Morgan upon the subject of decimal coinage. 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. In 1866 his life became clouded by the circumstances which led him to abandon the institution so long the scene of his labours. 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 Morgans sensitiveness on the subject of sectarian freedom; and, though his feelings were doubtless excessive, there is no doubt that gloom was thrown over his life, intensified in 1867 by the loss of 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 the 18th of March 1871. An interesting and truthful sketch of his life will be found in the Monthly Notices of the Royal Astronomical Society for the 9th of February 1872, vol. xxii. p. 112, written by A. C. Ranyard, who says, He was the kindliest, as well as the most learned of men benignant to every one who approached him, never forgetting the claims which weakness has on strength. De Morgan left no published indications of his opinions on religious questions, in regard to which he was extremely reticent. He seldom or never entered a place of worship, and declared that he could not listen to a sermon, a circumstance perhaps due to the extremely strict religious discipline under which he was brought up. Nevertheless there is reason to believe that he VIII. 1 a was of a deeply religious disposition. Like M. Faraday and Sir I.  Newton he entertained a confident belief in Providence, founded not on any tenuous inference, but on personal feeling. His hope of a future life also was vivid to the last. George Boole George Booles father, John Boole (1779–1848), was a tradesman of limited means, but of studious character and active mind. Being especially interested in mathematical science and logic, the father gave his son his first lessons; but the extraordinary mathematical talents of George Boole did not manifest themselves in early life. At first, his favorite subject was classics. It was not until his successful establishment of a school at Lincoln, its removal to Waddington, and later his appointment in 1849 as the first professor of mathematics of then Queens College, Cork in Ireland (now University College Cork, where the library, underground lecture theatre complex and the Boole Centre for Research in Informatics are named in his honour) that his mathematical skills were fully realized. In 1855 he married Mary Everest (niece of George Everest), who later, as Mrs. Boole, wrote several useful educational works on her husbands principles. To the broader public Boole was known only as the author of numerous abstruse papers on mathematical topics, and of three or four distinct publications that have become standard works. His earliest published paper was the Researches in the theory of analytical transformations, with a special application to the reduction of the general equation of the second order. printed in the Cambridge Mathematical Journal in February 1840 , and it led to a friendship between Boole and D. F. Gregory, the editor of the journal, which lasted until the premature death of the latter in 1844. A long list of Booles memoirs and detached papers, both on logical and mathematical topics, are found in the Catalogue of Scientific Memoirs published by the Royal Society, and in the supplementary volume on Differential Equations, edited by Isaac Todhunter. To the Cambridge Mathematical Journal and its successor, the Cambridge and Dublin Mathematical Journal, Boole contributed twenty-two articles in all. In the third and fourth series of the Philosophical Magazine are found sixteen papers. The Royal Society printed six important memoirs in the Philosophical Transactions, and a few other memoirs are to be found in the Transactions of the Royal Society of Edinburgh and of the Royal Irish Academy, in the Bulletin de lAcademie de St-Petersbourg for 1862 (under the name G Boldt, vol. iv. pp. 198–215), and in Crelles Journal. Also included is a paper on the mathematical basis of logic, published in the Mechanics Magazine in 1848. The works of Boole are thus contained in about fifty scattered articles and a few separate publications. Only two systematic treatises on mathematical subjects were completed by Boole during his lifetime. The well-known Treatise on Differential Equations appeared in 1859, and was followed, the next year, by a Treatise on the Calculus of Finite Differences, designed to serve as a sequel to the former work. These treatises are valuable contributions to the important branches of mathematics in question. To a certain extent these works embody the more important discoveries of their author. In the sixteenth and seventeenth chapters of the Differential Equations we find, for instance, an account of the general symbolic method, the bold and skilful employment of which led to Booles chief discoveries, and of a general method in analysis, originally described in his famous memoir printed in the Philosophical Transactions for 1844. Boole was one of the most eminent of those who perceived that the symbols of operation could be separated from those of quantity and treated as distinct objects of calculation. His principal characteristic was perfect confidence in any result obtained by the treatment of symbols in accordance with their primary laws and conditions, and an almost unrivalled skill and power in tracing out these results. During the last few years of his life Boole was constantly engaged in extending his researches with the object of producing a second edition of his Differential Equations much more complete than the first edition, and part of his last vacation was spent in the libraries of the Royal Society and the British Museum; but this new edition was never completed. Even the manuscripts left at his death were so incomplete that Todhunter, into whose hands they were put, found it impossible to use them in the publication of a second edition of the original treatise, and printed them, in 1865, in a supplementary volume. With the exception of Augustus de Morgan, Boole was probably the first English mathematician since the time of John Wallis who had also written upon logic. His novel views of logical method were due to the same profound confidence in symbolic reasoning to which he had successfully trusted in mathematical investigation. Speculations concerning a calculus of reasoning had at different times occupied Booles thoughts, but it was not till the spring of 1847 that he put his ideas into the pamphlet called Mathematical Analysis of Logic. Boole afterwards regarded this as a hasty and imperfect exposition of his logical system, and he desired that his much larger work, An Investigation of the Laws of Thought, on Which are Founded the Mathematical Theories of Logic and Probabilities (1854), should alone be considered as containing a mature statement of his views. Nevertheless, there is a charm of originality about his earlier logical work that is easy to appreciate.