Augustus De Morgan
1.1 Childhood 1.2 University education 1.3 London University 1.4 Family 1.5 Retirement and death
2 Mathematical work
3 Spiritualism 4 Legacy 5 Selected writings 6 See also 7 Notes and references 8 Further reading 9 External links
Augustus De Morgan
There is a word in our language with which I shall not confuse this subject, both on account of the dishonourable use which is frequently made of it, as an imputation thrown by one sect upon another, and of the variety of significations attached to it. I shall use the word Anti-Deism to signify the opinion that there does not exist a Creator who made and sustains the Universe. — De Morgan 1838, p. 22
In 1823, at the age of sixteen, he entered Trinity College,
Cambridge, where he came under the influence of
Eliza (1801–1836) married Lewis Hensley, a surgeon, living in Bath. Augustus (1806–1871) George (1808–1890), a barrister-at-law who married Josephine, daughter of Vice Admiral Josiah Coghill, 3rd Baronet Coghill Campbell Greig (1811–1876), a surgeon at the Middlesex Hospital
In the autumn of 1837, he married Sophia Elizabeth Frend (1809-1892), 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. He and another like-minded alumnus conceived the idea of founding a mathematical society in London, where mathematical papers would be not only received (as by the Royal Society) but actually read and discussed. The first meeting was held in University College; De Morgan was the first president, his son the first secretary. It was the beginning of the London Mathematical Society. Retirement and death
Augustus De Morgan.
In 1866 the chair of mental philosophy in University College fell vacant. James Martineau, a Unitarian clergyman and professor of mental philosophy, was recommended formally by the Senate to the Council; but in the Council there were some who objected to a Unitarian clergyman, and others who objected to theistic philosophy. A layman of the school of Bain and Spencer was appointed. De Morgan considered that the old standard of religious neutrality had been hauled down, and forthwith resigned. He was now 60 years of age. His 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 called, 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 18 March 1871. Mathematical work 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 Sir William Hamilton, 9th Baronet (that is, his title was inherited), 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 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 name was William Rowan Hamilton, contributed to mathematics, especially geometric algebra, and first 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 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 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.
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 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
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
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
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
might mean 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 sign
placed between two symbols, as in
A = B
. It indicates that the two symbols have the same resulting meaning, by whatever different steps attained. That
, if quantities, are the same amount of quantity; that if operations, they are of the same effect, etc.
(), 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
... renders it necessary to state them separately":
a = 0 + a
= + a
= a + 0
= a − 0
= 1 × a
displaystyle =1times a
= × a
displaystyle =times a
= a × 1
displaystyle =atimes 1
= a ÷ 1
displaystyle =adiv 1
= 0 + 1 × a
displaystyle =0+1times a
Law of signs.
+ ( + a ) = + a ,
+ ( − a ) = − a ,
− ( + a ) = − a ,
− ( − a ) = + a ,
× ( × a ) = × a ,
displaystyle times (times a)=times a,
× ( ÷ a ) = ÷ a ,
displaystyle times (div a)=div a,
÷ ( × a ) = ÷ a ,
displaystyle div (times a)=div a,
÷ ( ÷ a ) = × a
displaystyle div (div a)=times a
a + b = b + a ,
a × b = b × a
displaystyle atimes b=btimes a
a ( b + c ) = a b + a c ,
a ( b − c ) = a b − a c ,
( b + c ) ÷ a = ( b ÷ a ) + ( c ÷ a ) ,
displaystyle (b+c)div a=(bdiv a)+(cdiv a),
( b − c ) ÷ a = ( b ÷ a ) − ( c ÷ a )
displaystyle (b-c)div a=(bdiv a)-(cdiv a)
= 1 ,
displaystyle a^ 0 =1,
= a ,
displaystyle a^ 1 =a,
( a × b
displaystyle (atimes b)^ c =a^ c times b^ c ,
b + c
displaystyle a^ b times a^ c =a^ b+c ,
b × c
displaystyle (a^ b )^ c =a^ btimes 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 ) , ( a b ) c = a ( b c )
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
displaystyle m^ n
is not equal to
displaystyle 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.[citation
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 at Dublin, and
, of the M's that are A's is
, and of the M's that are B's is
; then there are at least
( a + b − m )
A's that are B's. Suppose that the number of souls on board a steamer
was 1000, that 500 were in the saloon, and 700 were lost. It follows
of necessity, that at least 700 + 500 - 1000, that is, 200, saloon
passengers were lost. This single principle suffices to prove the
validity of all the Aristotelian moods. It is therefore a fundamental
principle in necessary reasoning.
Here then De Morgan had made a great advance by introducing
quantification of the terms. At that time Sir William Hamilton was
A great many individuals, ever since the rise of the mathematical method, have, each for himself, attacked its direct and indirect consequences. I shall call each of these persons a paradoxer, and his system a paradox. I use the word in the old sense: a paradox is something which is apart from general opinion, either in subject matter, method, or conclusion. Many of the things brought forward would now be called crotchets, which is the nearest word we have to old paradox. But there is this difference, that by calling a thing a crotchet we mean to speak lightly of it; which was not the necessary sense of paradox. Thus in the 16th century many spoke of the earth's motion as the paradox of Copernicus and held the ingenuity of that theory in very high esteem, and some I think who even inclined towards it. In the seventeenth century the deprivation of meaning took place, in England at least.
How can the sound paradoxer be distinguished from the false paradoxer? De Morgan supplies the following test:
The manner in which a paradoxer will show himself, as to sense or nonsense, will not depend upon what he maintains, but upon whether he has or has not made a sufficient knowledge of what has been done by others, especially as to the mode of doing it, a preliminary to inventing knowledge for himself... New knowledge, when to any purpose, must come by contemplation of old knowledge, in every matter which concerns thought; mechanical contrivance sometimes, not very often, escapes this rule. All the men who are now called discoverers, in every matter ruled by thought, have been men versed in the minds of their predecessors and learned in what had been before them. There is not one exception.
The Budget consists of a review of a large collection of paradoxical books which De Morgan had accumulated in his own library, partly by purchase at bookstands, partly from books sent to him for review, partly from books sent to him by the authors. He gives the following classification: squarers of the circle, trisectors of the angle, duplicators of the cube, constructors of perpetual motion, subverters of gravitation, stagnators of the earth, builders of the universe. You will still find specimens of all these classes in the New World and in the new century. De Morgan gives his personal knowledge of paradoxers.
I suspect that I know more of the English class than any man in Britain. I never kept any reckoning: but I know that one year with another? — and less of late years than in earlier time? — I have talked to more than five in each year, giving more than a hundred and fifty specimens. Of this I am sure, that it is my own fault if they have not been a thousand. Nobody knows how they swarm, except those to whom they naturally resort. They are in all ranks and occupations, of all ages and characters. They are very earnest people, and their purpose is bona fide, the dissemination of their paradoxes. A great many — the mass, indeed — are illiterate, and a great many waste their means, and are in or approaching penury. These discoverers despise one another.
A paradoxer to whom De Morgan paid the compliment which Achilles paid Hector — to drag him round the walls again and again — was James Smith, a successful merchant of Liverpool. He found
π = 3
displaystyle pi =3 tfrac 1 8
. His mode of reasoning was a curious caricature of the reductio ad absurdum of Euclid. He said let
π = 3
displaystyle pi =3 tfrac 1 8
, and then showed that on that supposition, every other value of
must be absurd. Consequently,
π = 3
displaystyle pi =3 tfrac 1 8
is the true value. The following is a specimen of De Morgan's dragging round the walls of Troy:
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. A mathematical snail! This cannot be the thing so called which regulates the striking of a clock; for it would mean that I am to make Mr. Smith sound the true time of day, which I would by no means undertake upon a clock that gains 19 seconds odd in every hour by false quadrative value of
. But he ventures to tell me that pebbles from the sling of simple truth and common sense will ultimately crack my shell, and put me hors de combat. The confusion of images is amusing: Goliath turning himself into a snail to avoid
π = 3
displaystyle pi =3 tfrac 1 8
and James Smith, Esq., of the Mersey Dock Board: and put hors de combat by pebbles from a sling. If Goliath had crept into a snail shell, David would have cracked the Philistine with his foot. There is something like modesty in the implication that the crack-shell pebble has not yet taken effect; it might have been thought that the slinger would by this time have been singing — And thrice [and one-eighth] I routed all my foes, And thrice [and one-eighth] I slew the slain.
In the region of pure mathematics, De Morgan could detect easily the
false from the true paradox; but he was not so proficient in the field
of physics. His father-in-law was a paradoxer, and his wife a
paradoxer; and in the opinion of the physical philosophers De Morgan
himself scarcely escaped. His wife wrote a book describing the
phenomena of spiritualism, table-rapping, table-turning, etc.; and De
Morgan wrote a preface in which he said that he knew some of the
asserted facts, believed others on testimony, but did not pretend to
know whether they were caused by spirits, or had some unknown and
unimagined origin. From this alternative he left out ordinary material
causes. Faraday delivered a lecture on Spiritualism, in which he laid
it down that in the investigation we ought to set out with the idea of
what is physically possible, or impossible; De Morgan did not believe
De Morgan discovered relation algebra in his Syllabus of a Proposed
System of Logic (1966: 208-46), first published in 1860. This algebra
was extended by
Charles Sanders Peirce
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.
John Beloff wrote that De Morgan was the first
notable scientist in Britain to take an interest in the study of
spiritualism and his studies had influenced the decision of William
Crookes to also study spiritualism. He also claims that De Morgan was
an atheist and that this debarred him from a position at
An Explanation of the Gnomonic Projection of the Sphere. London:
Elements of Trigonometry, and Trigonometrical Analysis. London: Taylor
& Walton. 1837a.
The Elements of Algebra. London: Taylor & Walton. 1837b.
An Essay on Probabilities, and Their Application to Life Contingencies
and Insurance Offices. London: Longman, Orme, Brown, Green &
The Elements of Arithmetic]. London: Taylor & Walton. 1840a.
First Notions of Logic, Preparatory to the Study of Geometry. London:
Taylor & Walton. 1840b.
The Differential and Integral Calculus. London: Baldwin. 1842.
1845. The Globes, Celestial and Terrestrial. London: Malby & Co.
1847. Formal Logic or The Calculus of Inference, Necessary and
Probable. London: Taylor & Walton.
Notes and references
^ The year of his birth may be found by solving a conundrum proposed by himself, "I was x years of age in the year x2 (He was 43 in 1849). The problem is indeterminate, but it is made strictly determinate by the century of its utterance and the limit to a man's life. Those born in 1722 (1764–42), 1892 (1936–44) and 1980 (2025–45) are similarly privileged.
^ "De Morgan". Random House Webster's Unabridged Dictionary.
^ De Morgan, (1838) Induction (mathematics), The Penny Cyclopedia.
^ Beloff 1997, p. 47.
^ De Morgan & De Morgan 1882, p. 393.
^ "De Morgan, Augustus (D823A)". A
Beloff, John (1997). Parapsychology: A Concise History. Palgrave
Macmillan. ISBN 978-0-312-17376-0. He seems an unlikely convert
considering that his atheistic views had debarred him from a position
De Morgan, A., 1966. Logic: On the Syllogism and Other Logical
Writings. Heath, P., ed. Routledge. A useful collection of De Morgan's
most important writings on logic.
De Morgan, Sophia Elizabeth (1882). Memoir of Augustus De Morgan.
London: Longmans, Green and Company.
Grattan-Guinness, Ivor (2000). The Search for Mathematical Roots
1870-1940. Princeton University Press.
Macfarlane, Alexander (1916). Lectures on Ten British Mathematicians
of the Nineteenth Century (PDF). New York: John Wiley and Sons.
Merrill, Daniel D. (1990).
Augustus De Morgan
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Augustus De Morgan
WorldCat Identities VIAF: 29575875 LCCN: n80057177 ISNI: 0000 0001 0883 2969 GND: 119057972 SUDOC: 067050832 BNF: cb12174090t (data) MGP: 112646 NLA: 35034274 NDL: 00521093 SN