v 6.20.00
20 Mar 2021
updated 8 Jul 2021

Antique Algebra & Higher Arithmetic

Over the years I’ve collected I daren’t say how many books with a view to using them as references when I got the opportunity of putting together a stout volume that would be titled something like “Things I wish I’d known earlier” or “Things I wish had been properly explained at the time”.

In fact several slightly slimmer volumes would have been necessary, separately covering maths, physics and physical chemistry – not textbooks, of course, there are far too many of those anyway, but elaborations upon interesting topics that had only been touched upon, or remedial accounts of important topics that had been rather disgracefully mangled, during my undergraduate and postgraduate education.

The webpages that follow are in the first category – elaborations of algebraic techniques, old or sometimes new, that relate to the theory of numbers in all sorts of different ways. They don’t necessarily dig deep, there’s Wikipedia for the nitty-gritty, but they hopefully bring things out of shadow into sunlight.

Space is also devoted to the Higher Arithmetic, a rather quaint name for the arithmetical topics that are regarded as too esoteric for even senior secondary school but not sufficiently highbrow for university study – continued fractions, for example.

And there are topics which, in my pre-internet era weren’t even covered by the chemical physics literature, such as Gaussian quadrature, numerical simulation and so on – it was considered infra dig to refer to such sordid details, essential though they are to serious computation. I had to work things out for myself, in conjunction with my brilliant postgraduate colleague Robert S Milligan, though now, 50 years on, it’s probably much more professional. And of course now there’s the internet …

The (mostly popular) books listed below contributed to a greater or lesser extent to the stuff I’ve committed to my website – don’t think I’ve read them all cover to cover! Sometimes they contradict one another, and sometimes a popular book can unlock a concept much better than any textbook – as for instance (for me) the significance of Euclid’s Lemma for a really neat proof of the Fundamental Theory of Arithmetic.

Here and there I’ve also quoted from websites as indicated in situ.

But don’t mistake me for an expert in these things. Hopefully, each essay will reach ‘critical clarity’ and enable the interested reader (myself also) to tackle more substantial accounts elsewhere.

Irving Adler, Magic House of Number, Signet 1957

Irving Adler, The New Mathematics, Dobson 1959

E T Bell, Men of Mathematics, Scientific Book Club, date unknown

E T Bell, Mathematics: Queen and Servant of Science, G Bell & Sons 1954

E T Bell, The Last Problem, Victor Gollancz 1962

Douglas St P Barnard, Adventures in Mathematics, Pelham Books 1965

Margaret E Bowman, Romance in Arithmetic, University of London Press 1961

Jamie Buchan, As Easy as Pi, Michael O’Mara Books 2009

F J Budden, Number Scales and Computers, Longmans Green 1967

David M Burton, Elementary Number Theory, William C Brown 1988

John Conway & Richard Guy, The Book of Numbers, Copernicus 1996

Tony Crilly, 50 Ideas You Really Need to Know about Maths, Quercus undated

Tobias Danzig, Number, The Language of Science, Allen & Unwin 1962

John Derbyshire, Prime Obsession, Plume 2004

John Derbyshire, Unknown Quantity, Joseph Henry Press 2006

Keith Devlin, The Man of Numbers, Bloomsbury Publishing 2011

Keith Devlin, The Millennium Problems, Granta Books 2004

William Dunham, Journey through Genius, John Wiley 1990

William Dunham, The Mathematical Universe, John Wiley 1994

W L Ferrar, Higher Algebra for Schools, Clarendon Press 1945

Graham Flegg, Numbers: Their History and Meaning, Penguin 1984

Jan Gullberg, Mathematics from the Birth of Numbers, W W Norton 1997

H S Hall & S R Knight, Elementary Algebra for Schools, Macmillan 1899

H S Hall & S R Knight, Higher Algebra, Macmillan 1964

G H Hardy & E M Wright, An Introduction to the Theory of Numbers, Oxford 1960

Sir Thomas L Heath, A Manual of Greek Mathematics, Dover 2003

Peter M Higgins, Numbers - A Very Short Introduction, OUP 2011

Lancelot Hogben, Mathematics for the Million, Merlin Press 1989

Adrian Jenkins, The Number File, Tarquin Books 2000

Cornelius Lanczos, Numbers without End, Oliver and Boyd 1968

Eli Maor, The Pythagorean Theorem, Princeton 2007

John McLeish, Number, Flamingo, London 1992

Joaquin Navarro, The Secrets of π, Everything is Mathematical 2010

A Page, Algebra, University of London Press, 1965

W J Reichman, The Fascination of Numbers, Methuen 1957

Dan Rockmore, Stalking the Riemann Hypothesis, Jonathan Cape 2005

Evelyn B Rosenthal, Understanding the New Maths, Souvenir Press 1966

Karl Sabbagh, Dr Riemann’s Zeros, Atlantic Books 2002

Marcus du Sautoy, The Music of the Primes, Fourth Estate 2003

Charles Smith, A Treatise on Algebra, Macmillan 1946

Ian Stewart, Professor Stewart’s Incredible Numbers, Profile Books 2015

L F Taylor, Numbers, Faber & Faber 1970

I Todhunter, Theory of Equations, Macmillan 1875

Alistair Macintosh Wilson, The Infinite in the Finite, Oxford 1995

James Wood, Elements of Algebra, Cambridge 1848

You may well be disappointed, albeit only slightly, that there are as yet no links to the actual essays, the principal reason being that though many of them are substantially complete in MS Word 2003 (!), I’m still uncertain as to how best to present them on-line. There are several possible formats – html, pdf or even jpg, for example, but they each have serious drawbacks.

It could be alleged that this is very much a ‘scissors and paste’ construct, a bogus way of displaying second-hand expertise, with a smattering of commentary to hold it all together – at best, a conjoining of one man’s wit with other men’s wisdom. Or in the words of Montaigne, a garland of other men’s flowers and nothing is mine but the cord that binds them. But I believe that (in Dr Johnson’s words) I have touched nothing that I have not adorned to at least some slight extent.


All Ideas Are Second-Hand: Mark Twain’s Magnificent Letter to Helen Keller About the Myth of Originality

“The kernel, the soul — let us go further and say the substance, the bulk, the actual and valuable material of all human utterances — is plagiarism.”


The combinatorial nature of creativity is something I think about a great deal, so this 1903 letter Mark Twain wrote to his friend Helen Keller, found in Mark Twain’s Letters, Vol. 2 of 2, makes me nod with the manic indefatigability of a dashboard bobble-head dog. In this excerpt, Twain addresses some plagiarism charges that had been made against Keller some 11 years prior, when her short story The Frost King was found to be strikingly similar to Margaret Canby’s Frost Fairies.

Keller was acquitted after an investigation, but the incident stuck with Twain and prompted him to pen the following passionate words more than a decade later, which articulate just about everything I believe to be true of combinatorial creativity and the myth of originality:

“Oh, dear me, how unspeakably funny and owlishly idiotic and grotesque was that ‘plagiarism’ farce! As if there was much of anything in any human utterance, oral or written, except plagiarism! The kernel, the soul — let us go further and say the substance, the bulk, the actual and valuable material of all human utterances — is plagiarism. For substantially all ideas are second-hand, consciously and unconsciously drawn from a million outside sources, and daily used by the garnerer with a pride and satisfaction born of the superstition that he originated them; whereas there is not a rag of originality about them anywhere except the little discoloration they get from his mental and moral calibre and his temperament, and which is revealed in characteristics of phrasing. When a great orator makes a great speech you are listening to ten centuries and ten thousand men — but we call it his speech, and really some exceedingly small portion of it is his. But not enough to signify. It is merely a Waterloo. It is Wellington’s battle, in some degree, and we call it his; but there are others that contributed. It takes a thousand men to invent a telegraph, or a steam engine, or a phonograph, or a telephone or any other important thing — and the last man gets the credit and we forget the others. He added his little mite — that is all he did. These object lessons should teach us that ninety-nine parts of all things that proceed from the intellect are plagiarisms, pure and simple; and the lesson ought to make us modest. But nothing can do that.”

Steve Jobs, of course, knew this when he famously proclaimed that “creativity is just connecting things” — and Kirby Ferguson reminds us that Jobs didn’t technically invent any of the things that made him into a cultural icon, he merely perfected them to a point of genius. Still, this fear of unoriginality — and, at its extreme, plagiarism — plagues the creative ego like no other malady. No one has countered this paradox more eloquently and succinctly than Salvador Dalí: “Those who do not want to imitate anything, produce nothing.”