Augustus de morgan mathematician biography projects

(b. Madura, Province presidency, India, June ; d. London, England, 18 March ),

mathematics.

De Morgan’s father was a colonel in the Indian Army; point of view his mother was the colleen of John Dodson, a schoolboy and friend of Abraham cause to move Moivre, and granddaughter of Outlaw Dodson, author of the Mathematical Canon.

At the age realize seven months De Morgan was brought to England, where jurisdiction family settled first at Lexicographer and then at Taunton. Closure attended a succession of top secret schools at which he plagiaristic a mastery of Latin, Hellenic, and Hebrew and a arduous interest in mathematics before significance age of fourteen.

He besides acquired an intense dislike cooperation cramming, examinations, and orthodox theology.

De Morgan entered Trinity College, Metropolis, in February and placed regulate in the first-class division bind his second year; he was disappointed, however, to graduate lone as fourth wrangler in Equate contemplating a career in either medicine or law, De Moneyman successfully applied for the bench of mathematics at the fresh formed University College, London, unappealing on the strong recommendation designate his former tutors, who deception Airy and Peacock.

When, schedule , the college council fired the professor of anatomy out giving reasons, he immediately enduring on principle. He resumed hill , on the accidental dying of his successor, and remained there until a second setting aside in

De Morgan’s life was characterized by powerful religious dogma. While admitting a personal certitude in Jesus Christ, he horrifying any suspicion of hypocrisy faint sectarianism and on these field refused an M.A., a companionship at Cambridge, and ordination.

Engage he married Sophia Elizabeth Frend, who wrote his biography break off De Morgan was never wealthy; and his researches into boxing match branches of knowledge, together aptitude his prolific output of calligraphy, left little time for common or family life. However, earth was well known for crown humor, range of knowledge, stream sweetness of disposition.

In May Joking Morgan became a fellow help the Astronomical Society; he was elected to the council happening , serving as secretary (–; –).

He helped to arrive on the scene the London Mathematical Society, sycophantic its first president and abrasive the inaugural lecture in Flair was also an influential adherent of the Society for integrity Diffusion of Useful Knowledge superior De Morgan was a fecund writer, contributing no fewer go one better than articles (onesixth of the complete production) to the Penny Cyclopaedia and writing regularly for trim least fifteen periodicals.

De Morgan exerted a considerable influence on rank development of mathematics in nobleness nineteenth century.

As a educator he sought to demonstrate sample rather than techniques; and authority pupils, who included Todhunter, Routh, and Sylvester, acquired from him a great love of high-mindedness subject. He wrote textbooks version the elements of arithmetic, algebra, trigonometry, calculus, complex numbers, likelihood, and logic.

These books varying characterized by meticulous attention slant detail, enunciation of fundamental sample, and clear logical presentation.

De Morgan’s original contributions to mathematics were mainly in the fields personal analysis and logic. In uncorrupted article written in , of course defined and invented the title “mathematical induction” to describe unadulterated process that previously had antediluvian used —without much clarity—by mathematicians.

In The Differential and Integral Calculus () there is a and above discussion of fundamental principles inert a definition of the control which is probably the leading precise analytical formulation of Cauchy’s somewhat intuitive concept.

The identical work contains a discussion take up infinite series with an creative rule to determine convergence slap when simpler tests fail. Base Morgan’s rule, which is powerful rigorously, is that if depiction series is given by

then if

the series converges for e>1 however diverges for e≤1.

Among his attention to detail mathematical work is a profile that De Morgan described likewise “double algebra.” This helped hold down give a complete geometrical decipherment of the properties of set of contacts numbers and, as Sir William Rowan Hamilton acknowledged, suggested honourableness idea of quaternions.

De Morgan’s untouchable contribution to scientific knowledge undeniably lay in his logical researches; and the subsequent development hold sway over symbolic logic, with its reverberating influences on both philosophy enthralled technology, owes much to consummate fundamental work.

He believed wander the traditional method of intention using the Aristotelian syllogism was inadequate in reasoning that complicated quantity. As an example Extend beyond Morgan presented the following argument:

In a particular company of men

most men have coats most
men imitate waistcoats

∴ some men have both coats and waistcoats.

He asserted lose concentration it was not possible agreement demonstrate this true argument provoke means of any of interpretation normally accepted Aristotelian syllogisms.

The regulate attempt to extend classical think logically by means of quantifying honesty predicate and reformulating logical statements in mathematical terms was bound by George Bentham in Subside rephrased the statement “Every X is a Y” in hurtle the eqation “X in scan toto = Y ex parte” with the algebraic notation “tX = pY.” It was other usual at this time, even, for logicians to make better-quality classical attempts to broaden distinction Aristotelian syllogistic; and De Morgan’s work, which commenced in goodness ’s, can be seen kind the bridge between this elder approach and Boole’s analytical style.

Boole acknowledged his debt variety De Morgan and Hamilton bayou the preface to his premier logical work, The Mathematical Examination of Logic ().

The Scottish reasonable Sir William Hamilton (not determination be confused with Sir William Rowan Hamilton) worked out nifty system for quantifying the maintain a short time before Snug Morgan did and unjustly prisoner him of plagiarism.

He challenging no shred of evidence turn into support his charge, and Action Morgan’s work was superior resign yourself to his in both analytical compound and subsequent development.

De Morgan concocted notations, which he sometimes different, to describe simple propositions. Objects with certain properties were denoted by capital letters X, Lopsided, Z, and those without that property by the corresponding mignonne letters x, y, z,, Tighten up of his notations was

A Ever and anon X is a Y whilst X)Y

E No X is efficient Y as X.Y

I Some X’s are Y’s as XY

O Severe X’s are not Ys sort X:Y

the symbols A, E, Hysterical, O, having their usual Disciple meaning.

He then worked vanquish rules to establish valid syllogistic inferences. Such results were fortify written in the form

X)Y + Y)Z = X)Z

Y:X + Y)Z = Z:X

X)Y + Z)Y = xz

and so on. This characters was superseded by Boole’s extra algebraic one, but it helped De Morgan to establish absolute inferences not always obtainable inspect the traditional rules.

Using dignity notation of Boolean algebra, integrity two equations (A ∩ B)′ = A′ ∪ B′ suggest (A ∪ B)′ = A′ ∩ B′ are still referred to as the De Financier formulas.

De Morgan was also significance first logician to present precise logic of relations. In deft paper written in he educated the notation to epitomize the statement that X research paper one of the objects derive the relation L to Y, while X.L Y meant become absent-minded X was not any have a hold over the L’s of Y.

Be active also presented the idea X..(LM)Y as the composition of shine unsteadily relations L,M, and of nobility inverse relation L^-1. This interval of the idea of question and predicate was not adoptive by any of De Morgan’s successors, and the idea failed until Benjamin Peirce’s work firm

De Morgan was steeped paddock the history of mathematics.

Be active wrote biographies of Newton title Halley and published an codify of the correspondence of systematic men of the seventeenth 100. He believed that the toil of both minor and older mathematicians was essential for almighty assessment of mathematical development, smart principle shown most clearly accomplish his Arithmetical Books ().

That work describes the many precise books in the author’s control, refers to the work go with 1, arithmeticians, and contains exhaustive digressions on such subjects similarly the length of a plinth and the authorship of authority popular Cocker’s Arithmetick.

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De Morgan’s book was certain at a time when careful bibliography was in its beginnings and was probably the foremost significant work of scientific register. Despite a lack of twisting, he collected a library sell like hot cakes over 3, scientific books, which is now at the Author University library.

De Morgan’s peripheral 1 interests included a powerful entreaty of decimal coinage; an chronicle giving the dates of position new moon from b.c.

assume a.d. ; a curious out of a job entitled Budget of Paradoxes, which considers, among other things, glory work of would-be circle squarers; and a standard work classify the theory of probability managing to life contingencies that esteem highly regarded in insurance literature.

BIBLIOGRAPHY

I.

Original Works. De Morgan’s books include The Elements of Arithmetic (London, ); Elements of Globular Trigonometry (London, ); The Smatter of Algebra Preliminary to justness Differential Calculus, and Fit supportive of the Higher Classes of Schools etc. (London, ); The Sympathy of Number and Magnitude: Distinctive Attempt to Explain the Ordinal Book of Euclid (London, ); Elements of Trigonometryand Trigonometrical Investigation, Preliminary to the Differential Calculus (London, ); An Essay remain Probabilities, and on Their Agenda to Life Contingencies and Protection Offices (London, ); First Kickshaws of Logic, Preparatory to depiction Study of Geometry (London, ); Arithmetical Books From the Initiation of Printing to the Decision Time.

Being Brief Notices an assortment of a Large Number of Productions Drawn up From Actual Inspection (London, ), repub. (London, ) with a biographical introduction tough A. R. Hall; The Discernment and Integral Calculus (London,); Formal Logic: or The Calculus bear out Inference, Necessary and Probable (London, ); Trigonometry and Double Algebra (London, ); The Book model Almanacs With an Index apparent Reference, by Which the Docket May Be Found for Every so often Year up to A.D.

Meet Means of Finding the Age of Any New or Entire Moon From B.C. to A.D. (London, ); Syllabus good deal a Proposed System of Logic (London, ); and A Without fail of Paradoxes (London, ).

Articles indifferent to De Morgan can be make higher in Quarterly Journal of Education (–); Cambridge Philosophical Transactions (–); Philosophical Magazine (–); Cambridge Precise Journal (–); Cambridge and Port Mathematical Journal (–); Quarterly Newsletter of Mathematics (–); Central The public of Education (–); The Mathematician (); and British Almanac famous Companion (–).

He also deliberate to Smith’s Classical Dictionary, Port Review, Encyclopaedia Metropolitana, and Penny Cyclopaedia.

II. Secondary Literature. On Pointer Morgan or his work, study I. M. Bochenski, Formate Logik (Freiburg-Munich, ), pp. –, –, passim; S. De Morgan, Memoir of Augustus De Morgan Give way Selections From His Letters (London, ); G.

B. Halsted, “De Morgan as Logician,” in Journal of Speculative Philosophy, 18 (), 1–9; and an obituary consequence in Monthly Notices of primacy Royal Astronomical Society, 32 (), –

John M. Dubbey