v 6.20.00
20 Mar 2021
updated 20 Mar 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

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

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

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

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

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

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

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

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.