Augustus De Morgan

Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician and logician. He is best known for De Morgan's laws, relating logical conjunction, disjunction, and negation, and for coining the term "mathematical induction", the underlying principles of which he formalized. De Morgan's contributions to logic are heavily used in many branches of mathematics, including set theory and probability theory, as well as other related fields such as computer science.

Childhood
Augustus De Morgan was born in Madurai, in the Carnatic region of India, in 1806. His father was Lieutenant-Colonel John De Morgan (1772–1816), who held various appointments in the service of the East India Company, and his mother, Elizabeth (née Dodson, 1776–1856), was the daughter of John Dodson and granddaughter of James Dodson, who computed a table of anti-logarithms (inverse logarithms). Augustus De Morgan became blind in one eye within a few months of his birth. His family moved to England when Augustus was seven months old. As his father and grandfather had both been born in India, De Morgan used to say that he was neither English nor Scottish nor Irish, but a Briton "unattached," using the technical term applied to an undergraduate of Oxford or Cambridge who was not a member of any one of the colleges.

When De Morgan was ten years old, his father died. His mathematical talents went unnoticed until he was fourteen when a family friend discovered him making an elaborate drawing of a figure from one of Euclid's works with a ruler and compasses. He received his secondary education from Mr. Parsons, a fellow of Oriel College, Oxford, who preferred classics to mathematics.

Education
In 1823, at the age of sixteen, De Morgan enrolled in Trinity College, Cambridge, where his teachers and tutors included George Peacock, William Whewell, George Biddell Airy, H. Parr Hamilton, and John Philips Higman. Both Peacock and Whewell would influence De Morgan's selection of algebra and logic for further research.

He placed fourth in the Mathematical Tripos, which entitled him to the degree of Bachelor of Arts. To take the higher degree of Master of Arts and thereby become eligible for a fellowship, he was required to pass a theological test. Despite having been brought up in the Church of England, De Morgan strongly objected to submitting to such a test. Unable to progress in academia, De Morgan entered Lincoln's Inn to pursue law.

London University, 1827–1831
The London University (now known as University College London) was founded in 1826 as a secular alternative to Oxford and Cambridge; Catholics, Jews, and dissenters could enter as students and hold positions. Prior to opening in 1828, the University advertised 24 vacancies for professorship, two in mathematics, to which De Morgan applied.

De Morgan was appointed Professor of Mathematics on 23 February 1828. The Council of the London University had failed to recruit Charles Babbage and John Herschel to the position. Ultimately the search committee, steered by founder Lord Brougham, Olinthus Gregory, and Henry Warburton, selected De Morgan from a field of at least 31 candidates including Dionysius Lardner, Peter Nicholson, John Radford Young, Henry Moseley, John Herapath, Thomas Hewitt Key, William Ritchie, and John Walker.

De Morgan's work during this period focused on mathematical instruction: His first publication was The Elements of Algebra (1828), a translation of a French textbook by, followed by Elements of Arithmetic (1830), a widely used and long-lived textbook, and The Study and Difficulties of Mathematics (1831), a discourse on mathematical education.

Following a series of squabbles between the faculty, including De Morgan, and the administration, in particular the Warden, Leonard Horner, a dispute arose over the handling of medical student protests calling for the removal of the Professor of Anatomy, Granville Sharp Pattison, on the grounds of incompetence. While De Morgan and others argued that students should have no influence in the matter, the University bowed to student pressure and dismissed Pattison. De Morgan resigned on 24 July 1831, followed by Professors George Long and Friedrich August Rosen.

The Society for the Diffusion of Useful Knowledge
In 1826 Lord Brougham, one of the founders of London University, founded the Society for the Diffusion of Useful Knowledge (SDUK) with the goal of promoting self-education and improving the moral character of the middle- and working- classes through cheap and accessible publications. De Morgan became involved with the SDUK in March 1827; his unpublished manuscript Elements of Statics for the society may have played a role in his appointment to London University. One of its most voluminous and effective writers, De Morgan published several books with SDUK: On the Study and Difficulties of Mathematics (1831), Elementary Illustrations of the Differential and Integral Calculus (1832), The Elements of Spherical Trigonometry (1834), Examples of the Processes of Arithmetic and Algebra (1835), An Explanation of the Gnomic projection of the sphere (1836), The Differential and Integral Calculus (1842), and The Globes Celestial and Terrestrial (1845), as well as over 700 articles in the Penny Cyclopedia and contributions to the Quarterly Journal of Education, the Gallery of Portraits, and the Companion to the British Almanac.

Private tutor
Following his first resignation from London University, De Morgan started his work as a private tutor. One of his early students was Jacob Waley. He would tutor Ada Lovelace from 1840 through 1842, primarily via correspondence.

Actuary
De Morgan's great-grandfather, grandfather, and father-in-law were all actuaries; not surprisingly, De Morgan also worked as a consulting actuary for various life assurance firms, including the Family Endowment Assurance Office, the Albert Life Assurance Office, and the Alliance Assurance Company. He published several articles on actuarial subjects as well as the book An Essay on Probabilities and Their Application to Life Contingencies and Insurance Offices. However his most notable work as an actuary is his promotion of the work of Benjamin Gompertz, whose "law of mortality" was both under-appreciated and plagiarized.

Royal Astronomical Society
De Morgan became involved with the Astronomical Society of London in 1928. He would be appointed honorary secretary in 1931, the year in which it received its Royal Charter and became the Royal Astronomical Society. He would continue as secretary for 18 years and remain actively involved in the Society for 30 years.

London University, 1836–1866
In 1836, De Morgan's replacement as Professor of Mathematics, George J. P. White, drowned; De Morgan was convinced to return and reinstated. That same year the London University was renamed University College and, together with King's College, was made an affiliate of the newly created University of London.

De Morgan was a highly successful mathematics teacher. For over 30 years his courses covered a full curriculum, from Euclid through the calculus of variations, with his classes often exceeding 100 students. His approach integrated lectures, reading, problem sets, personal instruction, and extensive course notes. He disliked rote learning and viewed mathematics education as learning to reason and core to a liberal education. Several of his students went on to become mathematicians, most notably James Joseph Sylvester, and some of them, Edward Routh and Isaac Todhunter, well known educators themselves. Many of his non-mathematician students rated him highly; William Stanley Jevons described De Morgan as "unrivalled" as a teacher. Jevons, heavily influenced by De Morgan, would go on to do independent work in logic and become best known for the development of the theory of utility as part of the so-called Marginal Revolution.

In 1866, the Chair of Mental Philosophy and Logic at University College fell vacant and James Martineau was recommended formally by the Senate to the Council. The Council, at the urging of George Grote, rejected Martineau on the grounds that he was a Unitarian clergyman and instead appointed a layman, George Croom Robertson. De Morgan argued that the founding principle of religious neutrality had been abandoned and immediately resigned.

Ramchundra and Indian mathematics
In 1850 De Morgan received a book from John Elliot Drinkwater Bethune, A Treatise on Problems of Maxima and Minima, written and self-published by the self-taught Indian mathematician Ramchundra. De Morgan was so struck by the work that he entered into correspondence with Ramchundra and arranged for the book's re-publication in London in 1859, targeting a European audience; De Morgan's preface surveyed classical Indian mathematical thought and urged a contemporary return of Indian mathematics: "On examining this work I saw in it, not merely merit worthy of encouragement, but merit of a peculiar kind, the encouragement of which, as it appeared to me, was likely to promote native effort towards the restoration of the native mind in India."

The influence of classical Indian logic on De Morgan's own work on logic has been speculated upon. Mary Boole, claimed a profound influence—via her uncle George Everest—of Indian thought in general and Indian logic, in particular, on her husband George Boole, as well as on De Morgan:"Think what must have been the effect of the intense Hinduizing of three such men as Babbage, De Morgan, and George Boole on the mathematical atmosphere of 1830–65. What share had it in generating the vector analysis and the mathematics by which investigations in physical science are now conducted?"

London Mathematical Society
Arthur Cowper Ranyard and George Campbell De Morgan, De Morgan's son, conceived the idea of founding a mathematical society in London, where mathematical papers would be not only received (as by the Royal Society) but also read and discussed. The first meeting of the London Mathematical Society was held at University College in 1865. De Morgan was the first president and his son was the first secretary. The earliest members included Benjamin Gompertz, De Morgan's personal friend and fellow actuary, William Stanley Jevons and James Joseph Sylvester, De Morgan's former students, Thomas Archer Hirst, De Morgan's colleague, and mathematicians William Kingdom Clifford and Arthur Cayley.

Family
Augustus was one of seven children, only four of whom survived to adulthood. These siblings were Eliza (1801–1836), who married Lewis Hensley, a surgeon living in Bath; George (1808–1890), a barrister-at-law who married Josephine, daughter of Vice Admiral Josiah Coghill, 3rd Baronet Coghill; and Campbell Greig (1811–1876), a surgeon at the Middlesex Hospital.

When De Morgan came to live in London, he found a congenial friend in William Frend. Both were arithmeticians and actuaries, and their religious views were somewhat similar, although their mathematical views differed on account of Frend's rejection of the use of negative numbers. Frend lived in what was then a suburb of London, in a country house formerly occupied by Daniel Defoe and Isaac Watts. De Morgan, with his flute, was a welcome visitor.

In the autumn of 1837, De Morgan married Sophia Elizabeth Frend (1809–1892), the eldest daughter of William Frend (1757–1841), and Sarah Blackburne (1779–?), a granddaughter of Francis Blackburne (1705–1787), Archdeacon of Cleveland.

De Morgan had three sons and four daughters, including fairytale author Mary De Morgan. His eldest son was the potter William De Morgan. His second son, George, acquired distinction in mathematics at University College and the University of London.

Personality
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 offered to confer on him the honorary degree of LL. D.; he declined the honour as a misnomer. He humorously described himself using the Latin phrase 'Homo paucarum literarum ' (man of few letters), reflecting his modesty about his extensive contributions to mathematics and logic.

He disliked the provinces outside London, and while his family enjoyed the seaside and men of science were having a good time at a meeting of the British Association in the country, he remained in the hot and dusty libraries of the metropolis. He said that he felt like Socrates, who declared that the farther he was from Athens, the farther he was 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.

Religious views
Despite a strict Church of England upbringing De Morgan was publicly a non-comformist, at some personal cost: His refusal to conform debarred him from further advancement at Cambridge; his marriage was without Church ceremony; and on several occasions he fought with the University College administration to maintain religious neutrality, eventually resigning over the issue. In private De Morgan was a dissenter: He married into a Unitarian family, where his essentially Christian deist interpretations of scripture were welcome. Later in life he would lean more deist and join Martineau's Free Christian Union.

De Morgan was on occasion accused of atheism which he dismissed as sectarianism. In his will De Morgan would write "I commend my future with hope and confidence to Almighty God; to God the Father of our Lord Jesus Christ, whom I believe in my heart to be the Son of God, but whom I have not confessed with my lips, because in my time such confession has always been the way up in the world."

Retirement and death


At age 60, De Morgan's pupils secured him a pension of £500 p.a., but misfortunes followed. Two years later, his son George—the "younger Bernoulli," as Augustus loved to hear him call him, in allusion to the eminent father-and-son mathematicians of that name—died. This blow was followed by the death of a daughter. Five years after his resignation from University College, De Morgan died of nervous prostration on March 18, 1871.

Mathematics
De Morgan is best known for his pioneering contributions to mathematical logic, specifically algebraic logic, and, to a lesser extent, for his contributions to the beginnings of abstract algebra.

Mathematical logic
De Morgan's contributions to logic are two-fold. Firstly, before De Morgan there was no mathematical logic&mdash;logic, including formal logic, was the domain of philosophers; De Morgan was the first to make formal logic a mathematical subject. Secondly, De Morgan would develop the calculus of relations, essentially abstracting logic via the application of algebraic principles.

De Morgan's first original paper on logic, "On the structure of the syllogism", appeared in the Transactions of the Cambridge Philosophical Society in 1846. The paper describes a mathematical system that formalizes Aristotelian logic, specifically the syllogism. While the rules De Morgan defines, including the eponymous De Morgan's laws, are straightforward, the formalism is significant: it represented the first serious instance of mathematical logic, which would come to pervade the field of logic, and presaged logic programming. The subsequent dispute with the philosopher Sir William Stirling Hamilton over the "quantification of the predicate" referred to in De Morgan's paper would lead George Boole to write the pamphlet Mathematical Anaysis of Logic (1847). De Morgan elaborated upon his initial paper in the book Formal Logic, or the Calculus of Inference, Necessary and Probable (1847), published the same week as Boole's pamphlet and overshadowed by it.

De Morgan developed the calculus of relations in Syllabus of a Proposed System of Logic (1860). He showed that reasoning with syllogisms could be replaced with the composition of relations. The calculus was described as the logic of relatives by Charles Sanders Peirce, who admired De Morgan and met him shortly before his death.

A close study of De Morgan's contributions to algebraic logic has been recommended: "Any serious attempt to study the contemporary work of Tarski or Birkhoff should begin with a serious study of the most significant founders of their field, especially Boole, De Morgan, Pierce and Schröder". In fact, a theorem articulated by De Morgan in 1860 was later expressed by Schrŏder in his textbook on binary relations, and is now commonly called Schröder rules.

Commenting on De Morgan's contributions, C. I. Lewis wrote, "His originality in the invention of new logical forms, his ready wit, his pat illustrations, and clarity and liveliness of his writing did yeoman service in breaking down the prejudice against the introduction of 'mathematical' methods into logic".

Abstract algebra
In 1844, De Morgan, in response to W. R. Hamilton's invention of quaternions the previous year, articulated the notion of an abstract algebra: "Inventing a distinct system of unit-symbols, and investigating or assigning relations which define their mode of action on each other".

Works
De Morgan was a prolific writer; an incomplete list of his works occupies 15 pages of his memoirs. While most of De Morgan's mathematical writing is educational in nature, consisting of various textbooks, it is for his pioneering contributions to logic, and, to a lesser extent, algebra for which he is best known. His best known work on logic is the book Formal Logic, published in 1847. The best presentation of his view of algebra is found in a volume entitled Trigonometry and Double Algebra, published in 1849. De Morgan was also a well known popularizer of science and mathematics; he contributed over 600 articles to the Penny Cyclopedia, ranging from Abacus to Young, Thomas. His most unusual work is A Budget of Paradoxes, a compilation of his writing, mostly book reviews, for The Athenæum Journal.

Correspondence
De Morgan was a brilliant and witty writer, whether as a controversialist or as a correspondent. In his time, there flourished two Sir William Hamiltons who have often been conflated. One was William Hamilton, a Scotsman, professor of logic and metaphysics at the University of Edinburgh; the other was a knight (that is, won the title), an Irishman, professor of astronomy at the University of Dublin. "Be it known 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 bit 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 the second discoverer of a known theorem."

The correspondence of De Morgan with W. R. Hamilton, the mathematician, extended over twenty-four years; it contains discussions not only of mathematical matters but also of subjects of general interest. It is marked by geniality on the part of Hamilton and by wit on the part of De Morgan. The following 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 good English to use the same word in two different senses in one sentence, particularly when there is usage. Incongruity of language is no bull, for it expresses meaning. But incongruity of ideas (as in the case of the Irishman who was pulling up the rope, and finding it did not finish, cried out that somebody had cut off the other end of it) is the genuine bull."

Trigonometry and Double Algebra
De Morgan's work entitled Trigonometry and Double Algebra consists of two parts, the former of which is a treatise on trigonometry and the latter a treatise on generalized algebra, which he called "double algebra". The first stage in the development of algebra is arithmetic, where only natural numbers and symbols of operations such as $x^{2}$, $+$, etc. are used. The next stage is universal arithmetic, where letters appear instead of numbers so as to denote numbers universally, and the processes are conducted without knowing the values of the symbols. Let $x$ and $a$ denote any natural numbers. An expression such as $×$ may still be impossible, so in universal arithmetic there is always a proviso, provided the operation is possible. The third stage is single algebra, where the symbol may denote a quantity forwards or a quantity backwards and is adequately represented by segments on a straight line passing through an origin. Negative quantities are then no longer impossible; they are represented by the backward segment. But an impossibility still remains in the latter part of such an expression as $a − b$, which arises in the solution of the quadratic equation. The fourth stage is double algebra. The algebraic symbol denotes, in general, a segment of a line in a given plane. It is a double symbol because it involves two specifications, namely, length and direction, and $a + b√−1$ is interpreted as denoting a quadrant. The expression $√−1$ then represents a line in the plane having an abscissa $b$ and an ordinate $a$. Argand and Warren have carried double algebra so far, but they were unable to interpret on this theory such an expression as $a + b√−1$. De Morgan attempted it by reducing such an expression to the form $e^{a√−1}$, and he considered that he had shown that it could always be so reduced. The remarkable fact is that this double algebra satisfies all the fundamental laws above enumerated, and as every apparently impossible combination of symbols has been interpreted, it looks like the complete form of algebra. In Chapter 6, he introduced hyperbolic functions and discussed the connection between common and hyperbolic trigonometry.

If the above theory is true, the next stage of development ought to be triple algebra, and if $b + q√−1$ truly represents a line in a given plane, it ought to be possible to find a third term which that, added to the above, would represent a line in space. Argand and some others guessed that it was $a + b√−1$, although this contradicts the truth established by Euler that $a + b√−1 + c√−1^$. 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 quadruple algebra, when the axis of the plane is made variable. And this gives the answer to 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 remain 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 about the theory of symbolic algebra, De Morgan proceeds to give an inventory of the fundamental symbols of algebra, and also an inventory of the laws of algebra. The symbols are $$0$$, $$1$$, $$+$$, $$-$$, $$\times$$, $$\div$$, $$$$, and letters; these only, all others are derived. As De Morgan explains, the last of these symbols represents writing a latter expression in superscript over and after a former. His inventory of the fundamental laws is expressed under fourteen heads, but some of them are merely definitions. The preceding list of symbols is the matter under the first of these heads. The laws proper may be reduced to the following, which, as he admits, are not all independent of one another, "but the unsymmetrical character of the exponential operation, and the want of the connecting process of $$+$$ and $$\times$$ ... renders it necessary to state them separately":


 * 1) Identity laws. $$a = 0+a = 1 \times a$$, $$ = a+0 = a-0 = a \times 1 = a \div 1$$, $$ = 0 + 1 \times a $$
 * 2) Law of signs. $$+(+a) = +a,$$ $$+(-a) = -a,$$ $$-(+a) = -a,$$ $$-(-a) = +a,$$ $$\times (\times a) = \times a,$$ $$\times (\div a) = \div a,$$ $$\div (\times a) = \div a,$$ $$\div (\div a) = \times a$$
 * 3) Commutative law. $$ a + b = b + a,$$ $$a \times b = b \times a$$
 * 4) Distributive law. $$a(b + c) = ab + ac,$$ $$a(b - c) = ab - ac,$$ $$(b + c) \div a = (b \div a) + (c \div a),$$ $$(b - c) \div a = (b \div a) - (c \div a)$$
 * 5) Index laws. $$a^0=1,$$ $$a^1=a,$$ $$(a \times b)^c = a^c \times b^c,$$ $$a^b \times a^c = a^{b+c},$$ $$(a^b)^c = a^ {b \times c}$$

De Morgan professes to give a complete inventory of the laws which the symbols of algebra must obey, for he says, "Any system of symbols which obeys these rules and no others—except they be formed by combination of these rules—and which uses the preceding symbols and no others—except they be new symbols invented in abbreviation of combinations of these symbols—is symbolic algebra". From his point of view, none of the above principles are rules; they are formal laws, that is, arbitrarily chosen relations to which the algebraic symbols must be subject. He does not mention the law, which had already been pointed out by Gregory, namely, $$(a+b)+c = a+(b+c), (ab)c = a(bc)$$ and to which was afterwards given the name Law of association. If the commutative law fails, the associative may hold good; but not vice versa. It is an unfortunate thing for the symbolist or formalist that in universal arithmetic $$m^n$$ is not equal to $$n^m$$; for then the commutative law would have full scope. Why does he not give it full scope? Because the foundations of algebra are, after all, real not formal, material not symbolic. To the formalists the index operations are exceedingly refractory, in consequence of which some take no account of them, but relegate them to applied mathematics. To give an inventory of the laws which the symbols of algebra must obey is an impossible task, and reminds one not a little of the task of those philosophers who attempt to give an inventory of the a priori knowledge of the mind.

George Peacock's theory of algebra was much improved by D. F. Gregory, a younger member of the Cambridge School, who laid stress not on the permanence of equivalent forms but on the permanence of certain formal laws. This new theory of algebra as the science of symbols and of their laws of combination was carried to its logical issue by De Morgan, and his doctrine on the subject 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 may remark that he practically abandons it when he takes up the subject of infinite series. De Morgan's theory is stated 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 most important that the student should bear in mind that, with one exception, no word nor sign of 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 any one were to assert that $+$ and $-$ might mean reward and punishment, and $A$, $B$, $C$, 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 sign $$=$$ placed between two symbols, as in $$A = B$$. It indicates that the two symbols have the same resulting meaning, by whatever different steps are taken. That $$A$$ and $$B$$, if quantities, are the same amount of quantity; that if operations, they are of the same effect, etc.

Formal Logic
When the study of mathematics revived at the University of Cambridge, so did the study of logic. The moving spirit was Whewell, the Master of Trinity College, whose principal writings were a History of the Inductive Sciences and Philosophy of the Inductive Sciences. Doubtless De Morgan was influenced in his logical investigations by Whewell, but other influential contemporaries were Sir William Rowan Hamilton in Dublin and George Boole in Cork. De Morgan's work, Formal Logic, published in 1847, is principally remarkable for his development of the numerically definite syllogism. The followers of Aristotle say that from two particular propositions, such as Some M's are A's and Some M's are B's, nothing follows of necessity about the relation of the A's and B's. But they go further and say that in order that any relation between the A's and B's may follow of necessity, the middle term must be taken universally in one of the premises. De Morgan pointed out that from Most M's are A's and Most M's are B's, it follows by necessity that some A's are B's, and he formulated the numerically definite syllogism, which puts this principle in exact quantitative form. Suppose that the number of the M's is $$m$$, of the M's that are A's is $$a$$, and of the M's that are B's is $$b$$; then there are at least $$(a + b - m)$$ A's that are B's. Suppose that there are 1000 passengers on a ship, that 500 are in the ship's bar, and 700 of the passengers are lost. It necessarily follows that at least (700 + 500) – 1000, that is, 200, of the passengers in the bar are lost. This single principle proves the validity of all the Aristotelian moods. It is therefore a fundamental principle in necessary reasoning.

By then, De Morgan had made a great advance by introducing quantification of the terms. At that time, Sir William Hamilton was teaching in Edinburgh a doctrine of the quantification of the predicate, and a correspondence sprang up. However, De Morgan soon perceived that Hamilton's quantification was of a different character; that it meant, for example, substituting the two forms. The whole of A is the whole of B, and the whole of A is a part of B for the Aristotelian form. All A's are B's. Hamilton thought that he had placed the keystone in the Aristotelian arch, as he phrased it. Although it must have been a curious arch that could stand for 2000 years without a keystone, as a consequence, he had no room for De Morgan's innovations. He accused De Morgan of plagiarism, and the controversy raged for years in the columns of the Athenæum and in the publications of the two writers.

A Budget of Paradoxes
Published posthumously in 1872, A Budget of Paradoxes is a compilation of De Morgan's column of the same name for the Athenæum, consisting mostly book reviews and focusing on so-called paradoxers, also referred to as pseudomaths (a De Morgan neologism) and pseudoscientists.

The pseudomaths De Morgan describes are mostly circle-squarers, such as Thomas Baxter, cube-duplicators, and angle-trisectors. One such angle-trisector was James Sabben, whose work received a one-line review from De Morgan: "'The consequence of years of intense thought': very likely, and very sad."

Another pseudomath identified by De Morgan was James Smith, a successful merchant of Liverpool, who claimed that $$\pi = 3 \tfrac{1}{8}$$. De Morgan writes:

"Mr. Smith continues to write me long letters, to which he hints that I am to answer. In his last of 31 closely written sides of note paper, he informs me, with reference to my obstinate silence, that though I think myself and am thought by others to be a mathematical Goliath, I have resolved to play the mathematical snail, and keep within my shell... But he ventures to tell me that pebbles from the sling of simple truth and common sense will ultimately crack my shell..."

Among the many pseudoscientific ideas De Morgan discredits are Alfred Wilks Drayson's expanding Earth theory and Samuel Rowbotham's Zetetic Astronomy, or the flat Earth theory.

In his discussion of calculations of $$\pi$$, De Morgan discusses at length Buffon's approximation and his own results using the method.

De Morgan gives space to non-technical subjects in Budget as well, religion in particular. De Morgan gives a favorable review of Godfrey Higgins' Anacalypsis and provides several anecdotes about the views of great mathematicians on religion, notably Laplace and Euler.

De Morgan frequently displays humor in Budget, including various anagrams such as, "Great Gun, do us a sum!" (="Augustus De Morgan"), The Astronomer's Drinking Song, and the poem Siphonaptera. Budget was well-received but hard to categorize.

Spiritualism
De Morgan later in his life became interested in the phenomenon of spiritualism. Initially intrigued by clairvoyance, he later conducted paranormal investigations with the American medium Maria Hayden, the results of which appear in the book From Matter to Spirit: The Result of Ten Years Experience in Spirit Manifestations (1863), written by Sophia De Morgan and published anonymously to avoid repercussions.

Sophia may have been a convinced spiritualist, but De Morgan was neither a believer nor a skeptic. Instead, his viewpoint was that the methodology of the physical sciences does not automatically exclude psychic phenomena and that such phenomena may be explainable in time by the possible existence of natural forces that physicists have not yet identified. In the preface to From Matter to Spirit (1863), De Morgan writes: "Thinking it very likely that the universe may contain a few agencies – say half a million – about which no man knows anything, I can not but suspect that a small proportion of these agencies – say five thousand – may be severally competent to the production of all the [spiritualist] phenomena, or may be quite up to the task among them. The physical explanations which I have seen are easy, but miserably insufficient: the spiritualist hypothesis is sufficient, but ponderously difficult. Time and thought will decide, the second asking the first for more results of trial."

De Morgan was one of the first notable scientists in Britain to take an interest in the study of spiritualism, influencing William Crookes to also study spiritualism.

Legacy
The headquarters of the London Mathematical Society are called De Morgan House, and the top prize awarded by the Society is the De Morgan Medal.

The student society of the Mathematics Department of University College London is called the Augustus De Morgan Society.

De Morgan's extensive library of mathematical and scientific works, many historical, was acquired by Samuel Jones-Loyd for the University of London and is now part of the Senate House Libraries collection.

The lunar crater De Morgan is named after him.