# Augustus De Morgan

### Quick Info

Madura, Madras Presidency, India (now Madurai, Tamil Nadu, India)

London, England

**Augustus De Morgan**became the first professor of mathematics at University College London and made important contributions to English mathematics.

### Biography

**Augustus De Morgan**'s father, John De Morgan (5 October 1771 - 27 November 1816), was a Lieutenant-Colonel in the Madras Native Infantry. He was born and served in India in the 22 Dragoon Guards and married Elizabeth Dodson in 1798 at Colombo, Ceylon. Elizabeth was the daughter of John Dodson of the Custom House, London and the great-granddaughter of James Dodson (1705-1757) who published

*The Anti-Logarithmic Canon. Being a table of numbers consisting of eleven places of figures, corresponding to all Logarithms under 100,000, with an Introduction containing a short account of Logarithms*in 1742. John and Elizabeth De Morgan had seven children: John Augustus De Morgan (born 16 May 1799 and died when the

*Prince of Wales*was wrecked on its passage back from India in 1804); James De Morgan (also died in the 1804 wreck of the

*Prince of Wales*); Eliza De Morgan (born 27 September 1801); Georgina De Morgan (born March 1805 and died in 1812); Augustus De Morgan (born 27 June 1806, the subject of this biography); George De Morgan (born 15 July 1808, who became a barrister and died in 1890); and Campbell Grieg De Morgan (born 22 November 1811, who became a famous surgeon and died 12 April 1876).

Augustus lost the sight of his right eye shortly after birth when both eyes were affected with Indian "sore eye". One of his eyes was saved but he became blind in one eye. He was baptised on 20 October 1806 at Fort St George, Madras, India. When seven months old, he returned to England with his parents, and his sisters Eliza and Georgina. The family sailed to England in the

*Duchess of Gordon*, one of many ships in a convoy, and settled in Worcester. Augustus's father returned to India on his own in 1808, but returned to England in 1810. They lived at Appledore, then at Bideford, then at Barnstaple, all in Devon. In 1912 the family settled in Taunton in Somerset. John De Morgan returned to Madras in India but in 1816 became ill with a liver problem and died in St Helena on a return voyage to England. Augustus was 10 years old when his father died but, in a list of teachers made by him in later life, he gave his father as his first teacher.

De Morgan's schooling began in Barnstaple where he was taught reading and writing by Miss Williams, then in Taunton where, 1813-14, Mrs Poole taught him reading, writing and arithmetic and in the next couple of years the Rev J Fenner taught him Greek and Latin. Later in Blandford he was taught by the Rev T Keynes, then at Taunton, he was taught Latin, Greek, Euclidean geometry and algebra by the Rev H Barker. Finally he attended Mr Parsons' school, at Redland, near Bristol, where he studied from age fourteen to sixteen and a half. At Mr Parsons' school, De Morgan did not excel and, because of his physical disability [23]:-

... he did not join in the sports of other boys, and he was even made the victim of cruel practical jokes by some schoolfellows.For further details of De Morgan's time at Mr Parsons' school, see THIS LINK.

What we have not mentioned when giving details of De Morgan's education is his religious education. This, however, was highly significant since the strict training he received put him off the Church, although he remained a committed Christian. His mother wanted him to become an Evangelical Minister in the Church and put pressure on him to study at university with this aim. His schoolmaster, Mr Parsons, put pressure on him to study classics at university, but De Morgan's love was mathematics.

De Morgan entered Trinity College Cambridge in February 1823 at the age of 16 where he was taught mathematics by George Peacock and William Whewell - the three became lifelong friends. His College tutor was J P Higman, and he also attended lectures by George B Airy, Henry Coddington (1798-1845), and Henry Parr Hamilton (1794-1880). Although De Morgan's undergraduate career was successful, nevertheless, he did not shine in the way one might expect and there must have been a number of reasons for this. His mother put pressure on him about religion which gave him difficulties. He probably devoted too much time to his study of Classics, certainly in his first years, and his health was poor at times. He had the habit of studying all through the night, then getting up very late which may have contributed to his health problems. He was also unsure of where his studies should lead and in his final couple of years he thought seriously about a medical career. We noted above that his younger brother Campbell Grieg De Morgan did follow a medical career.

Perhaps De Morgan's greatest relaxation while a student was in playing the flute which he did to a high standard. Many of his friends would love to listen to his flute playing and would ask him to play.

He received his B.A. in 1827, being Fourth Wrangler in the Mathematical tripos. Henry Percy Gordon (1806-1876) was Senior Wrangler; he had a career in law. Thomas Turner (1804-1883) was Second Wrangler and First Smith's Prizeman. Turner also had a career in law but was an early fellow of the Royal Astronomical Society and had a lifelong interest in astronomy. Anthony Cleasby (1804-1879) was Third Wrangler; he also had a career in law. Although the three above De Morgan were undoubtedly extremely able, as their subsequent careers showed, nevertheless it seems certain that they lacked De Morgan's mathematical abilities. Certainly another factor here was De Morgan's dislike of the tripos type examination where cramming was the key to success rather than demonstrating originality [15]:-

The place of the youthful wrangler, though it failed to declare his real power or the exceptional aptitude of his mind for mathematical study, would, however, have been sufficient to have secured for him a fellowship, and he, no doubt, would have found a congenial field of labour within the walls of his university, if his conscientious scruples had not prevented his signing the tests which at that time were required from those who took up their degree of M.A. as well as from all Fellows of Colleges.Because a theological test was required for the M.A., 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 M.A. In 1826 he returned to his home in London and, despite having doubts that his conscience would make him a poor lawyer, he entered Lincoln's Inn to study for the Bar. He made it clear where his real interests were in one of his letters [7]:-

You seem to fancy that I was going to the Bar from choice. The fact is, that of all the professions which are called learned, the Bar was the most open to me; but my choice will be to keep to the sciences as long as they will feed me. I am very glad that I can sleep without the chance of dreaming that I see an "Indenture of Five Parts," or some such matter, held up between me and the 'Mecanique Celeste', knowing all the time that the dream must come true.In 1827 (at the age of 21) he applied for the chair of mathematics in the newly founded London University and, despite having no mathematical publications, he was appointed. On 23 February 1828, De Morgan became the first professor of mathematics at the London University; he gave his inaugural lecture

*On the study of mathematics*. In this lecture [27]:-

... De Morgan described mathematics as the deductive study of self-evident laws or axioms concerning clear and distinct ideas. ... he praised Locke's 'Essay Concerning Human Understanding' and claimed: "It is notorious that the first ideas which any human being receives are derived either from the figure or number of the objects which surround him. From the appearances of the material world, certain distinct notions are gathered, which though their prototypes have no real existence in nature, are the clearest and most definite which our minds contain."Sophia De Morgan writes [7]:-

This lecture 'On the Study of Mathematics' takes a much wider view of that study, and its effects upon the mind, than its title alone would imply. It is an essay upon the progress of knowledge, the need of knowledge, the right of everyone to as much knowledge as can be given to him, and the place in mental development which the culture of the reasoning power ought to hold. It is not only a discourse upon mental education, but upon mind itself.Teaching was, De Morgan said, the best way to learn a subject. He [15]:-

... began to teach himself to better purpose than he had been taught, as does every man who is not a fool, when he begins to teach others, let his former teachers be what they may.In 1828 De Morgan published

*The Elements of Algebra*, his English translation of the first three chapters of

*Élémens d'algèbre*by Pierre Louis Marie Bourdon (1779-1854). This book was "designed for the use of students in the University of London." In it, De Morgan writes (dated August 1828):-

The following translation has been prepared for the use of such students in the University of London as may not be able to read French, or do not desire to pursue their algebraical studies further than Equations of the Second Degree. The original work, in the opinion of the translator, is particularly well adapted for elementary instruction, on account of the care which is taken to deduce every rule from first principles, and to distinguish between the results of convention and those of demonstration. A translation of the whole would have been attempted, but or the consideration that at present every one who is desirous of attaining a considerable degree of mathematical knowledge must become acquainted with the French language; and it is to such only that the whole book would be necessary.De Morgan is very keen to distinguish between a theorem and a problem and on the first page he added the following "translator's note":-

*The first is a theorem, the second a problem.*

*The greater of two numbers is equal to half their sum added to half their difference.**What two numbers are those whose sum is 20 and whose difference is 10.*

Observe, that by a negative quantity is only meant a quantity to be subtracted; and by such an expression asThe summer of 1829 was spent in Paris where he met Jean Hachette, Jean-Baptiste Biot among others. He exchanged several letters with Hachette over the next few years until Hachette's death in 1834. In 1830 De Morgan published

$-a - a = -2a$,

is meant that subtraction of a from any number twice following gives the same result as the subtraction of $2a$ once. To guard against erroneous ideas concerning the meaning of the negative sign, the student should accustom himself to translate into common language such expressions as

5 - 8 = -3

which means, that the addition of 5 and the subtraction of 8, performed one after the other, is equivalent to the subtraction of 3 ...

*Elements of Arithmetic*. He wrote:-

This little work is an attempt to give the young student the common rules of Arithmetic, accompanied by the reasoning to which he must habituate his mind before he can make progress in any science.De Morgan was to resign his chair, on a matter of principle, is 1831. Some biographies of De Morgan state that he resigned because a fellow professor was dismissed. Although this is true, the reasons are somewhat more complex and involve the whole way in which the London University was governed. That the professors could be dismissed without good cause by a governing body which had little academic expertise was something that De Morgan felt strongly about. He wrote (see [7]):-

I might speak from experience, of the nature of the arithmetical knowledge which most youths acquire before they commence the study of geometry and Algebra. But as almost all agree in opinion, that this science ought not to be, as it is in this country, degraded into a mass of rules learned by rote, one half of which are of no use but in commercial business, and rarely even there, I will proceed to make a remark on the manner in which this book should be studied.

In order to avoid the generalities of algebraic language, which the mind of a beginner cannot grasp, it is necessary to confine each demonstration to one particular case; that is, to show, on some particular numbers, those truths which, in Algebra, are asserted of all at once, by means of letters to stand for numbers. From the case which is chosen, a rule is drawn which is assumed to hold good always. This reasoning is not strictly logical; but it must be recollected, that the student has it in his power to convince himself of the universal truth of what is stated, by employing different numbers from those used in the text, in every demonstration. This is what I recommend him to do: if he omits this exercise, he does not give the subject a fair trail.

In compliance with the wish expressed by you when I had the honour of an interview with you, I lay before you the views which I entertain on a subject most essentially connected with the welfare of the University, viz., the situation which the Professors ought to hold in the establishment. This question is of the highest importance, inasmuch as upon the manner in which it shall be settled depends the order of education and merit which will be found among the Professors in future, and the estimation in which they will be held by the public.These problems, which were there from his first appointment, came to a head with the dismissal of Granville Sharp Pattison (1791-1851), the first professor of anatomy at the London University.

In order to induce men of character to fill the chairs of the University, these latter must be rendered highly independent and respectable. No man who feels (rightly) for himself will face a class of pupils as long as there is anything in the character in which he appears before them to excite any feelings but those of the most entire respect. The pupils all know that there is a body in the University superior to the Professors; they should also know that this body respects the Professors, and that the fundamental laws of the institution will protect the Professor as long as he discharges his duty, as certainly as they will lead to his ejectment in case of misconduct or negligence. Unless the pupils are well assured of this they will look upon the situation of Professor as of very ambiguous respectability, and they will only be wrong inasmuch as there will be no ambiguity at all in the case.

For further details, including De Morgan's resignation letter, see THIS LINK.

After resigning, De Morgan moved from the family home in Guilford Street to 5 Upper Gower Street. There is an obvious question at this point, namely how did he support himself financially for five years without a job? It appears that he earned money by taking private pupils and by giving actuarial advice to various companies. The London University appointed George James Pelly White to succeed De Morgan as Professor of Mathematics. White was similar to De Morgan in having been a Trinity man with the same tutors and referees; in fact he stood out as clearly the best candidate.

Perhaps the most important work that De Morgan undertook during this period was his work for the Royal Astronomical Society. He had been elected a fellow on 9 May 1828 and served as secretary to the Society from 1831 to 1839 (again from 1847 to 1855) [5]:-

... this want of rapid publication of results was rendered less harmful by the excellent and fairly detailed summaries of all papers read, which now became a regular feature of the 'Monthly Notices'. ... there can be no doubt that De Morgan, who was Secretary from 1831-39, deserves a considerable share of the credit of this very useful part of the Society's publications. Throughout his life De Morgan continued to be warmly interested in the Society and was a regular attendant at the meetings. ... he firmly declined the office of President, which he did not think ought to be held by a man who was not an active worker in astronomy. ... His personal brilliance, his learning, at once extensive and minute, historical and modern, his hold on the best mathematics of the day, much in advance of his contemporaries, have made his name rather increase than diminish with the intervening decades. But in his relations to the Council it is his personal side that concerns us, that master passion for principle which was more than any reward or success for him.He was appointed to the chair again in 1836, after George White died in a boating accident, and held it until 1866 when he was to resign for a second time, again on a matter of principle.

For details of his 1836 appointment, see THIS LINK.

For details of his 1866 resignation, see THIS LINK.

De Morgan married Sophia Elizabeth Frend (1809-1892) on 3 August 1837. De Morgan had met Sophia ten years earlier through his friendship with her father William Frend who worked at the Nautical Almanac. Frend had published

*Principles of Algebra*(1796) with an appendix by Francis Maseres; Frend rejected the use of negative quantities. Because of his strong views, De Morgan did not want a Church wedding with the usual marriage ceremony so they were married in a Registry Office by the Rev Thomas Madge. The form of service omitted the 'duties of husbands and wives' part of the wedding service. Augustus and Sophia De Morgan had seven children: Elizabeth Alice De Morgan (born June 1838); William Frend De Morgan (born November 1839); George Campbell De Morgan (born October 1841); Edward I De Morgan (born June 1843); Anne Isabella De Morgan (born 11 February 1845); Helena Christiana De Morgan (born 20 March 1848); Mary Augusta De Morgan (born 24 February 1850).

In 1838 he defined and introduced the term 'mathematical induction' putting the 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 the London University, and that Society also published a famous work by De Morgan

*The Differential and Integral Calculus*(1836). In this he:-

... endeavoured to make limits the sole foundation of the science, without any aid whatsoever from the theory of series, or algebraical expressions.In 1849 he published

*Trigonometry and double algebra*in which he gave a geometric interpretation of complex numbers. He writes in the Preface:-

The work before the reader is entirely new, not being in any sense a second edition of that which I published on the same subject in 1837. It consists of two books. In the first, I have endeavoured to give the student who has a competent knowledge of arithmetic and algebra ... a view of trigonometry, as a branch of algebra and a constituent part of the foundation of higher mathematics. In the second, I have given n elementary view in its purely symbolic character, with the application of that geometrical basis of significance which affords explanation of ever symbol.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.

De Morgan corresponded with Charles Babbage and gave private tuition to Ada Lovelace who, it is claimed, wrote the first computer program for Babbage. He also corresponded with Hamilton and, like Hamilton attempted to extend double algebra to three dimensions. In a letter to Hamilton, De Morgan writes of his correspondence with Hamilton and William Hamilton. He writes:-

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, generalise it at a glance, bestow it thus generalised upon society at large, and make me the second discoverer of a known theorem.In 1864 he was a co-founder of the London Mathematical Society, suggesting its name, and became its first president. We quote, because of its relevance to this Archive, part of his President's Address of 16 January 1865 given at the 'First Meeting of the Society':-

I say that no art or science is a liberal art or a liberal science unless it is studied in connection with the mind of man in past times. It is astonishing how strangely mathematicians talk of the Mathematics, because they do not know the history of their subject. By asserting what they conceive to be facts they distort its history in this manner. There is in the idea of everyone some particular sequence of propositions, which he has in his own mind, and he imagines that the sequence exists in history; that his own order is the historical order in which the propositions have successively been evolved. The mathematician needs to know what the course of invention has been in the different branches of Mathematics; he wants to see Newton bringing out and evolving the Binomial Theorem by suggestion of the higher theorem which Wallis had already given. If he be to have his own researches guided in the way which will best lead him to success, he must have seen the curious ways in which the lower proposition has constantly been evolved from the higher.De Morgan's son George, a very able mathematician, became the first secretary of the London Mathematical Society. De Morgan was never a Fellow of the Royal Society of London as he refused to let his name be put forward. He also refused an honorary degree from the University of Edinburgh. He was described by Thomas Hirst thus:-

A dry dogmatic pedant I fear is Mr De Morgan, notwithstanding his unquestioned ability.Macfarlane remarks that [23]:-

... De Morgan considered himself a 'Briton unattached' neither English, Scottish, Welsh or Irish.He also writes [23]:-

He disliked the country 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 had no ideas or sympathies in common with the physical philosopher. His attitude was doubtless due to his physical infirmity, which prevented him from being either an observer or an experimenter. He never voted in an election, and he never visited the House of Commons, or the Tower, or Westminster Abbey.De Morgan was always interested in odd numerical facts and, writing in 1864, he noted that he had the distinction of being $x$ years old in the year $x^{2}$ (He was 43 in 1849). Anyone born in 1980 can claim the same distinction in 2025.

For details of De Morgan's final years, see THIS LINK.

Five days after his death, on 23 March 1871, his funeral was held and he was buried at All Souls, Kensal Green, Kensington and Chelsea, London.

### References (show)

- J M Dubbey, Biography in
*Dictionary of Scientific Biography*(New York 1970-1990). See THIS LINK. - Biography in
*Encyclopaedia Britannica.*http://www.britannica.com/biography/Augustus-De-Morgan - Obituary in
*The Times* - J M Curran,
*William Whewell, Augustus De Morgan, and the Nomenclature and Meaning of Victorian Induction*(Ph.D. Thesis, University of Oklahoma, 2014). - J L E Dreyer and H H Turner (eds.),
*History of the Royal Astronomical Society*(Royal Astronomical Society, London, 1923). - D D Merrill, Augustus De Morgan and the Logic of Relations (Kluwer Academic Publishers, Dordrecht-Boston-London, 1990).
- S E De Morgan,
*Memoir of Augustus De Morgan by his wife Sophia Elizabeth De Morgan*(London, 1882). - A C Rice,
*Augustus De Morgan and the development of university mathematics in London in the nineteenth century*(Ph.D. Thesis, Middlesex University, 2010). - Augustus De Morgan,
*Art UK*.

https://artuk.org/discover/artworks/augustus-de-morgan-18061871-248632 - N L Biggs, De Morgan on map colouring and the separation axiom,
*Arch. Hist. Exact Sci.***28**(2) (1983), 165-170. - N L Biggs, E K Lloyd and R J Wilson, C S Peirce and De Morgan on the four-colour conjecture,
*Historia Math.***4**(1977), 215-216. - S H Brown, The Life and Work of Augustus De Morgan,
*Applied Probability Trust*(2006), 4-9. - E V Cherkasova, Definition of transformation group in De Morgan's book 'On the foundation of algebra' (Russian),
*Voprosy Istor. Estestvoznan. i Tekhn.*(1) (1992), 90-92. - A Church, Review: Augustus De Morgan, On the Syllogism and other Logical Writings by Augustus De Morgan,
*Journal of Symbolic Logic***41**(2) (1976), 546-547. - Fellows deceased: Prof De Morgan,
*Monthly Notices of the Royal Astronomical Society***32**(1872), 112-118. - S Gandon, La théorie des rapports chez Augustus De Morgan,
*Revue d'Histoire des Sciences***62**(1) (2009), 285-311. - R C Gupta, Augustus De Morgan (1806-1871),
*Indian Journal of the History of Science***41**(4) (2006), 411. - G B Halsted, Biography: De Morgan,
*Amer. Math. Monthly***4**(1897), 1-5. - B S Hawkins, De Morgan, Victorian syllogistic and relational logic,
*Modern Logic***5**(2) (1995), 131-166. - B S Hawkins, A reassessment of Augustus De Morgan's logic of relations : a documentary reconstruction,
*Internat. Logic Rev.*(19-20) (1979), 32-61. - R Higgitt, Why I don't FRS my Tail: Augustus De Morgan and the Royal Society,
*Notes and Records of the Royal Society***60**(3) (2006), 253-259. - L M Laita, Influences on Boole's logic : the controversy between William Hamilton and Augustus De Morgan,
*Ann. of Sci.***36**(1) (1979), 45-65. - A Macfarlane, Augustus De Morgan, in
*Lectures on Ten British Mathematicians of the Nineteenth Century*(New York, 1916), 19-33. - D D Merrill, Augustus De Morgan's Boolean Algebra,
*History and Philosophy of Logic***26**(2) (2005), 75-91. - B H Neumann, Augustus De Morgan,
*Bull. London Math. Soc.***16**(1984), 575-589. - C Phillips, August De Morgan and the propagation of moral mathematics,
*Studies in History and Philosophy of Science Part A***36**(1) (2005), 105-133. - H M Pycior, Augustus De Morgan's algebraic work: the three stages,
*Isis***74**(272) (1983), 211-226. - A C Ranyard, Augustus De Morgan,
*Nature***18**(1) (1871), 409-410. - A Rice, Augustus De Morgan: Historian of science,
*History of Science***34**(1996), 201-240. - A Rice, Augustus De Morgan (1806-1871),
*The Mathematical Intelligencer***18**(3) (1996), 40-43. - A Rice, What makes a Great Mathematics Teacher? The case of Augustus De Morgan,
*Amer. Math. Monthly***106**(6) (1999), 534-552. - A Rice, Augustus De Morgan, in
*C C Heyde, E Seneta, P Crépel, S E Fienberg and J Gani (eds.), Statisticians of the Centuries*(Springer, New York,2001), 157-162. - J L Richards, Augustus De Morgan, the history of mathematics, and the foundations of algebra,
*Isis***78**(291) (1987), 7-30. - J L Richards, "This Compendious Language": Mathematics in the World of Augustus De Morgan,
*Isis***102**(3) (2011), 506-510. - C Simmons, Augustus De Morgan Behind the Scenes,
*The College Mathematics Journal***42**(1) (2011), 33-40. - C Simmons, Augustus De Morgan Behind the Scenes, in
*Mircea Pitici, The Best Writing on Mathematics 2012*(Princeton University Press, 2012), 186-197. - G C Smith, De Morgan and the laws of algebra,
*Centaurus***25**(1-2) (1981/82), 50-70. - G C Smith, De Morgan and the transition from infinitesimals to limits,
*Austral. Math. Soc. Gaz.***7**(2) (1980), 46-52.

### Additional Resources (show)

Other pages about Augustus De Morgan:

- Errors of the Royal Society
- De Morgan and the Royal Astronomical Society
- John Walsh's delusions by De Morgan
- De Morgan's Preface to Ramchundra's book
- Times obituary
- Multiple entries in
*The Mathematical Gazetteer of the British Isles*, - Hamilton and De Morgan discuss Buée's 1806 paper
- Heinz Klaus Strick biography
- De Morgan by his wife Sophia. Part 1
- De Morgan by his wife Sophia. Part 2
- De Morgan by his wife Sophia. Part 3
- De Morgan by his wife Sophia. Part 4

Other websites about Augustus De Morgan:

### Honours (show)

Honours awarded to Augustus De Morgan

### Cross-references (show)

- History Topics: A history of Pi
- History Topics: Extracts from Thomas Hirst's diary
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- Societies: London Mathematical Society
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- Other: Earliest Known Uses of Some of the Words of Mathematics (F)
- Other: Earliest Known Uses of Some of the Words of Mathematics (I)
- Other: Earliest Known Uses of Some of the Words of Mathematics (L)
- Other: Earliest Known Uses of Some of the Words of Mathematics (M)
- Other: Earliest Known Uses of Some of the Words of Mathematics (P)
- Other: Earliest Known Uses of Some of the Words of Mathematics (Q)
- Other: Earliest Known Uses of Some of the Words of Mathematics (R)
- Other: Earliest Known Uses of Some of the Words of Mathematics (S)
- Other: Earliest Known Uses of Some of the Words of Mathematics (T)
- Other: Earliest Known Uses of Some of the Words of Mathematics (U)
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Written by J J O'Connor and E F Robertson

Last Update July 2020

Last Update July 2020