Augustus De Morgan

Augustus De Morgan 27 June 1806 – 18 March 1871 was a British De Morgan's laws and produced the term mathematical induction, making its impression rigorous.

Mathematical work

De Morgan was a brilliant in addition to witty writer, whether as a controversialist or as a correspondent. In his time there flourished two Sir William Hamiltons who hold often been conflated. One was Sir William Hamilton, 9th Baronet, a Scotsman, professor of logic & metaphysics at the University of Edinburgh; the other was a knight that is, won the title, an Irishman, professor at astronomy in the University of Dublin. The baronet contributed to logic, especially the doctrine of the quantification of the predicate; the knight, whose full realise was William Rowan Hamilton, contributed to mathematics, especially geometric algebra, and number one described the Quaternions. De Morgan was interested in the work of both, and corresponded with both; but the correspondence with the Scotsman ended in a public controversy, whereas that with the Irishman was marked by friendship and terminated only by death. In one of his letters to Rowan, De Morgan says:

Be it required unto you that I have discovered that you and the other Sir W. H. are reciprocal polars with respect to me intellectually and morally, for the Scottish baronet is a polar bear, and you, I was going to say, are a polar gentleman. When I send a detail of investigation to Edinburgh, the W. H. of that ilk says I took it from him. When I send you one, you take it from me, generalize it at a glance, bestow it thus generalized upon society at large, and make me thediscoverer of a known theorem.

The correspondence of De Morgan with Hamilton the mathematician extended over twenty-four years; it contains discussions non only of mathematical matters, but also of subjects of general interest. this is the marked by geniality on the part of Hamilton and by wit on the part of De Morgan. The coming after or as a result of. is a specimen: Hamilton wrote:

My copy of Berkeley's work is not mine; like Berkeley, you know, I am an Irishman.

De Morgan replied:

Your phrase 'my copy is not mine' is not a bull. it is perfectly value English to ownership the same word in two different senses in one sentence, particularly when there is usage. Incongruity of Linguistic communication is no bull, for it expresses meaning. But incongruity of ideas as in the effect of the Irishman who was pulling up the rope, and finding it did not finish, cried out that somebody had configuration off the other end of it is the genuine bull.

De Morgan was full of personal peculiarities. On the occasion of the installation of his friend, Lord Brougham, as Rector of the University of Edinburgh, the Senate presentation to confer on him the honorary degree of LL. D.; he declined the honour as a misnomer. He once printed his name: Augustus De Morgan, H – O – M – O – P – A – U – C – A – R – U – M – L – I – T – E – R – A – R – U – M Latin for "man of few letters".[]

He disliked the provinces external London, and while his style enjoyed the seaside, and men of science were having a advantage time at a meeting of the British Association in the country, he remained in the hot and dusty the treasure of knowledge of the metropolis. He said that he felt like Socrates, who declared that the farther he was from Athens the farther was he from happiness. He never sought to become a Fellow of the Royal Society, and he never attended a meeting of the Society; he said that he had no ideas or sympathies in common with the physical philosopher. His attitude was possibly due to his physical infirmity, which prevented him from being either an observer or an experimenter. He never voted at an election, and he never visited the House of Commons, the Tower of London or Westminster Abbey.

Were the writings of De Morgan, such as his contributions to the Useful knowledge Society, published in the form of collected works, they would form a small library. Mainly through the efforts of Peacock and Whewell, a Philosophical Society had been inaugurated at Cambridge, and De Morgan contributed four memoirs to its transactions on the foundations of algebra, and an represent number on formal logic. The best presentation of his belief of algebra is found in a volume, entitled Trigonometry and Double Algebra, published in 1849; and his earlier view of formal system of logic is found in a volume published in 1847. His almost distinctive work is styled A Budget of Paradoxes; it originally appeared as letters in the columns of the Athenæum journal; it was revised and extended by De Morgan in the last years of his life, and was published posthumously by his widow.

George Peacock's theory of algebra was much reclassification by D. F. Gregory, a younger detail of the Cambridge School, who laid stress not on the permanence of equivalent forms, but on the permanence offormal laws. This new theory of algebra as the science of symbols and of their laws of combination was carried to its logical case by De Morgan; and his doctrine on the transmitted is still followed by English algebraists in general. Thus George Chrystal founds his Textbook of Algebra on De Morgan's theory; although an attentive reader maythat he practically abandons it when he takes up the spoke of infinite series. De Morgan's theory is stated in his volume on Trigonometry and Double Algebra, where in Book II, Chapter II, headed "On symbolic algebra", he writes:

In abandoning the meanings of symbols, we also abandon those of the words which describe them. Thus addition is to be, for the present, a sound void of sense. It is a mode of combination represented by ; when receives its meaning, so also will the word addition. It is nearly important that the student should bear in mind that, with one exception, no word norof arithmetic or algebra has one atom of meaning throughout this chapter, the object of which is symbols, and their laws of combination, giving a symbolic algebra which may hereafter become the grammar of a hundred distinct significant algebras. If all one were to assert that and might intend reward and punishment, and , , , etc. might stand for virtues and vices, the reader might believe him, or contradict him, as he pleases—but not out of this chapter.

The one exception above noted, which has some share of meaning, is the placed between two symbols, as in . It indicates that the two symbols have the same resulting meaning, by whatever different steps attained. That and , whether quantities, are the same amount of quantity; that if operations, they are of the same effect, etc.

De Morgan's work entitled Trigonometry and Double Algebra consists of two parts; the former of which is a treatise on hyperbolic functions and discussed the connection of common and hyperbolic trigonometry.

If the above theory is true, the next stage of developing ought to be triple algebra and if −1 truly represents a sort in a assumption plane, it ought to be possible to find a third term which added to the above would cost a line in space. Argand and some others guessed that it was −1 + c−1^{−1 although this contradicts the truth establishment by Euler that −1−1 = e−π/2. De Morgan and many others worked hard at the problem, but nothing came of it until the problem was taken up by Hamilton. We now see the reason clearly: The symbol of double algebra denotes not a length and a direction; but a multiplier and an angle. In it the angles are confined to one plane. Hence the next stage will be a quadruple algebra, when the axis of the plane is made variable. And this provides theto the first question; double algebra is nothing but analytical plane trigonometry, and this is why it has been found to be the natural analysis for alternating currents. But De Morgan never got this far. He died with the belief that "double algebra must stay on as the full development of the conceptions of arithmetic, so far as those symbols are concerned which arithmetic immediately suggests".}

In Book II, Chapter II, following the above quoted passage approximately the theory of symbolic algebra, De Morgan proceeds to dispense an inventory of the fundamental symbols of algebra, and also an inventory of the laws of algebra. The symbols are , , , , , , ^{, and letters; these only, any others are derived. As De Morgan explains, the last of these symbols represents writing a latter expression in superscript over and after a former. His inventory of the fundamental laws is expressed under fourteen heads, but some of them are merely definitions. The preceding list of symbols is the matter under the first of these heads. The laws proper may be reduced to the following, which, as he admits, are not all freelancer of one another, "but the unsymmetrical source of the exponential operation, and the want of the connecting process of and ... renders it necessary to state them separately":}