Re: [HM] Usage of the word abacus

David Fowler (david.fowler@warwick.ac.uk)
Fri, 20 Nov 1998 08:33:07 +0000

Julio Gonzalez Cabillon mentioned my interest back in December 1996 in the
words algorithm, algorism, algoristics, ... This was to prepare a note for
the 2nd ed of my book _The Mathematics of Plato's Academy_ to correct a
misunderstanding (perhaps, more accurately, a mistake!) in the first
edition. This note had to be short (to fit in the available space -- though
at a very late stage it was decided to reset the while book, so this became
irrelevant) and biased towards the interests of the book as a whole, so it
omits the detail that has been under discussion in the recent discussion on
the math-history list. In particular, it doesn't engage with the issue of
augrym, What I eventually arrived at is the following. (This omits italic,
a bit of Greek, formatting, etc.)

The first edition of this book attempted to reintroduce the word 'algorism'
to replace 'algorithm', a word castigated by the Oxford English Dictionary
(1933), s.v. algorism, as a one of the "many pseudo-etymological
perversions, including a recent algorithm in which [algorism] is learnedly
confused with the Greek arithmon, number". (The Supplement to the OED (in
vol. 1, 1972, incorporated into the OED second edition of 1989), admitted
the word and cited, as its first example, the first edition of Hardy &
Wright, ITN, (1938), a passage that was slightly changed in the second
edition of 1945.) In fact, the early meaning of 'algorism' is actually
'calculation with Hindu-Arabic numerals', as opposed to calculation with
the abacus. During the twelfth to fifteenth centuries, there were hundreds
of treatises whose titles or incipits were variations on De arte numerandi
algoristica, while the abacus treatises might be called Practica aritmetica
cum denariis proiectilibus or Algorithmus linealis. This last description
might be seen as a precursor of our modern use of 'algorithm'; in any case,
the word has an even more established and distinguished history than I had
thought. Its first indubitably modern use, as far as I know, - and what
must surely be its first significant mathematical use - was, in Latin, in
1684, in Leibniz' first publication on the calculus in Acta Eruditorum,
indeed in the first publication of the calculus, 'Nova Methodvs pro maximis
et minimis...': "Ex cognito hoc velut Algorithmo, ut ita dicam, calculi
hujus, quem voco differentialem,...". This short article of six pages was
translated by Joseph Raphson in 1715, in The Theory of Fluxions...., where
this passage is rendered "Now from this being known as the Algorithm, as I
may say of this Calculus, which I call differential, ..." (p.23). The word
was then taken up by Euler, for instance in his article 'De usu novi
algorithmi in problemate Pelliano solvendo' (and the 'Pell' here gave rise
to the misnomer described in Section 2.4(c), above), and its use was then
firmly established.

David Fowler