Algorism (augrim, augrym, algram, agram, algorithm), owes its name to
the accident that the first arithmetical treatise translated from the
Arabic happened to be one written by Al-Khowarazmi in the early ninth
century, “de numeris Indorum,” beginning in its Latin form “Dixit
Algorismi. . . .” The translation, of which only one MS. is known, was
made about 1120 by Adelard of Bath, who also wrote on the Abacus and
translated with a commentary Euclid from the Arabic. It is probable that
another version was made by Gerard of Cremona (1114-1187); the number of
important works that were not translated more than once from the Arabic
decreases every year with our knowledge of medieval texts. A few lines
of this translation, as copied by Halliwell, are given on p. 72, note 2.
Another translation still seems to have been made by Johannes
Hispalensis.
Algorism is distinguished from Abacist computation by recognising seven
rules, Addition, Subtraction, Duplation, Mediation, Multiplication,
Division, and Extraction of Roots, to which were afterwards added
Numeration and Progression. It is further distinguished by the use of
the zero, which enabled the computer to dispense with the columns of the
Abacus. It obviously employs a board with fine sand or wax, and later,
as a substitute, paper or parchment; slate and pencil were also used in
the fourteenth century, how much earlier is unknown.[5*] Algorism
quickly ousted the Abacus methods for all intricate calculations, being
simpler and more easily checked: in fact, the astronomical revival of
the twelfth and thirteenth centuries would have been impossible without
its aid.
[Footnote 5*: Slates are mentioned by Chaucer, and soon after
(1410) Prosdocimo de Beldamandi speaks of the use of a “lapis”
for making notes on by calculators.]
The number of Latin Algorisms still in manuscript is comparatively
large, but we are here only concerned with two--an Algorism in prose
attributed to Sacrobosco (John of Holywood) in the colophon of a Paris
manuscript, though this attribution is no longer regarded as conclusive,
and another in verse, most probably by Alexander de Villedieu (Villa
Dei). Alexander, who died in 1240, was teaching in Paris in 1209. His
verse treatise on the Calendar is dated 1200, and it is to that period
that his Algorism may be attributed; Sacrobosco died in 1256 and quotes
the verse Algorism. Several commentaries on Alexander’s verse treatise
were composed, from one of which our first tractate was translated, and
the text itself was from time to time enlarged, sections on proofs and
on mental arithmetic being added. We have no indication of the source on
which Alexander drew; it was most likely one of the translations of
Al-Khowarasmi, but he has also the Abacists in mind, as shewn by
preserving the use of differences in multiplication. His treatise, first
printed by Halliwell-Phillipps in his _Rara Mathematica_, is adapted for
use on a board covered with sand, a method almost universal in the
thirteenth century, as some passages in the algorism of that period
already quoted show: “Est et alius modus qui utitur apud Indos, et
doctor hujusmodi ipsos erat quidem nomine Algus. Et modus suus erat in
computando per quasdam figuras scribendo in pulvere. . . .” “Si
voluerimus depingere in pulvere predictos digitos secundum consuetudinem
algorismi . . .” “et sciendum est quod in nullo loco minutorum sive
secundorum . . . in pulvere debent scribi plusquam sexaginta.”