. I
(
ἄλγος
), meaning “pain”. Rather, it is a corruption of the name of the 9th century
Persian mathematician Mu
h
.
ammad ibn M
¯
us
¯
a al-Khw
¯
arizm
¯
ı.
1
Al-Khw
¯
arizm
¯
ı is
perhaps best known as the writer of the treatise Al-Kit
¯
ab al-mukhta
s
.
ar f
¯
ıh
¯
ıs
¯
ab
al-ğabr
wa’l-muq
¯
abala,
2
from which the modern word algebra derives. In a
different treatise, al-Khw
¯
arizm
¯
ı described the modern decimal system for writing
and manipulating numbers—in particular, the use of a small circle or
s
.
ifr to
represent a missing quantity—which had been developed in India several
centuries earlier. The methods described in Al-Kit
¯
ab, using either written figures
or counting stones, became known in English as algorism or augrym, and its
figures became known in English as ciphers.
Although both place-value notation and al-Khw
¯
arizm
¯
ı’s works were already
known by some European scholars, the “Hindu-Arabic” numeric system was
popularized in Europe by the medieval Italian mathematician and tradesman
Leonardo of Pisa, better known as Fibonacci. Thanks in part to his 1202 book
Liber Abaci,
3
written figures began to replace the counting table (then known
as an abacus) and finger arithmetic
4
as the preferred platform for calculation
5
in Europe in the 13th century—not because written decimal figures were any
easier to learn or use, but because they provided an audit trail. Ciphers became
common in Western Europe only with the advent of movable type, and truly
ubiquitous only after cheap paper became plentiful in the early 19th century.
Eventually the word algorism evolved into the modern algorithm, via folk
etymology from the Greek arithmos (and perhaps the previously mentioned
algos).
6
Thus, until very recently, the word algorithm referred exclusively
1
“Mohammad, father of Adbdulla, son of Moses, the Kw
¯
arizmian”. Kw
¯
arizm is an ancient
city, now called Khiva, in the Khorezm Province of Uzbekistan.
2
“The Compendious Book on Calculation by Completion and Balancing”
3
While it is tempting to translate the title Liber Abaci as “The Book of the Abacus”, a more
accurate translation is “The Book of Calculation”. Both before and after Fibonacci, the Italian
word abaco was used to describe anything related to numerical calculation—devices, methods,
schools, books, and so on—much in the same way that “computer science” is used today in
English, or as the Chinese phrase for “operations research” translates literally as “the study of
using counting rods”.
4
+ Reckoning with digits!
+
5
The word calculate derives from the Latin word calculus, meaning “small rock”, referring to
the stones on a counting table, or as Chaucer called them, augrym stones. In 440bc, Herodotus
wrote in his Histories (II.36) that “The Greeks write and calculate (
λογίζεσθαι ψήφοις
, literally
“reckon with pebbles”) from left to right; the Egyptians do the opposite. Yet they say that their
way of writing is toward the right, and the Greek way toward the left.” (Herodotus is strangely
silent on which end of the egg the Egyptians ate first.)
6
Some medieval sources claim that the Greek prefix “algo-” means “art” or “introduction”.
Others claim that algorithms were invented by a Greek philosopher, or a king of India, or perhaps
a king of Spain, named “Algus” or “Algor” or “Argus”. A few, possibly including Dante Alighieri,
even identified the inventor with the mythological Greek shipbuilder and eponymous argonaut.
It’s unclear whether any of these risible claims were intended to be historically accurate, or
merely mnemonic.
2