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Augustus de Morgan loved unusual and rare books, often spending all his spare money on a new volume. He was especially an admirer of the works of Charles Dickens. From his love of books, he developed and wrote about library cataloging systems, some of which are still in use today. Another of his loves was music, and he was an accomplished flutist.

De Morgan was also interested in the history of mathematics. He wrote biographies of Sir Isaac Newton and Edmund Halley. His wife wrote De Morgan's biography in 1882. His researches into all branches of knowledge and his prolific writing left him little time for social or family life, but he was well-known for his sense of humor.

DeMorgan pointed out that every science that has thrived, has thrived upon its own symbols, and that logic had not developed any symbols. This was what he set out to remedy. He realized the close relationship which existed between logic and pure mathematics, and believed these disciplines should be treated jointly and not separately. His studies in logic were of the highest value both in illuminating new areas and in encouraging others to follow in his footsteps.

He is perhaps best known as the creator of De Morgan's Laws:

NOT (A AND B) = (NOT A) OR (NOT B)

NOT (A OR B) = (NOT A) AND (NOT B)

These statements show that negating an AND makes it an OR and vice-versa. These laws have become some of the most frequently applied in modern proof theory and are widely used in software programming as well.

In 1838 he defined and introduced the term 'mathematical induction' putting a process that had been used without clarity on a rigorous basis. The term first appears in De Morgan's article Induction (Mathematics) in the Penny Cyclopedia. (Over the years he was to write 712 articles for the Penny Cyclopedia.) The Penny Cyclopedia was published by the Society for the Diffusion of Useful Knowledge, set up by the same reformers who founded London University, and that Society also published a famous work by De Morgan The Differential and Integral Calculus. In 1849 he published Trigonometry and double algebra in which he gave a geometric interpretation of complex numbers. He recognised the purely symbolic nature of algebra and he was aware of the existence of algebras other than ordinary algebra. He introduced De Morgan's laws and his greatest contribution is as a reformer of mathematical logic.

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 entered Trinity College Cambridge in 1823 at the age of 16 where he was taught by Peacock and Whewell - the three became lifelong friends. He received his BA but, because a theological test was required for the MA, something to which De Morgan strongly objected despite being a member of the Church of England, he could go no further at Cambridge being not eligible for a Fellowship without his MA. In 1826 he returned to his home in London and entered Lincoln's Inn to study for the Bar. In 1827 (at the age of 21) he applied for the chair of mathematics in the newly founded University College London and, despite having no mathematical publications, he was appointed.

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 George Biddell Airy.

Augustus DeMorgan was born in 1806 in Madras Province, India, where his father was employed by the East India Company. He received his early education in English private schools, which he hated. He had lost the use of one eye in infancy which made him shy and solitary and it exposed him to schoolboy pranks. One such prank was for a boy to sneak up on DeMorgan on his blind side, and holding a sharp-pointed penknife to his cheek, to suddenly call his name. When DeMorgan turned around he would be poked by the point of the knife on his face. DeMorgan did not allow other children to bully him and proved this by chasing down and beating up the boy who did this.

Augustus De Morgan was a brilliant mathematician and professor noted for his advances in algebra and logic. He was a friend of Charles Babbage (inventor of the Analytical Engine, forerunner of the modern computer) and he tutored Ada Lovelace, who was the poet Lord Byron's daughter and supposedly wrote the first computer program for Babbage's machine. De Morgan was prolific and wrote such mathematical texts as Elements of Arithmetic (1830), Penny Cyclopedia (1838) where he coined the term 'mathematical induction', Trigonometry and Double Algebra (1849), a geometric interpretation of complex numbers, and Formal Logic (1847), one of his most important works.