rites III III for 33, ICCOO and I. II. [tau]. [tau] for 1200,
I. O. VIII. IX for 1089, and I. IIII. IIII. [tau][tau][tau][tau] for the
square of 1200.

The period from the time of Gerbert until after the appearance of
Leonardo's monumental work may be called the period of the abacists. Even
for many years after the appearance early in the twelfth century of the
books explaining the Hindu art of reckoning, there was strife between the
abacists, the advocates of the abacus, and the algorists, those who favored
the new numerals. The words _cifra_ and _algorismus cifra_ were used with a
somewhat derisive significance, indicative of absolute uselessness, as
indeed the zero is useless on an abacus in which the value of any unit is
given by the column which it occupies.[475] So Gautier de Coincy
(1177-1236) in a work on the miracles of Mary says:

A horned beast, a sheep,
An algorismus-cipher,
Is a priest, who on such a feast day
Does not celebrate the holy Mother.[476]

So the abacus held the field for a long time, even against the new algorism
employing the new numerals. {121} Geoffrey Chaucer[477] describes in _The
Miller's Tale_ the clerk with

"His Almageste and bokes grete and smale,
His astrelabie, longinge for his art,
His augrim-stones layen faire apart
On shelves couched at his beddes heed."

So, too, in Chaucer's explanation of the astrolabe,[478] written for his
son Lewis, the number of degrees is expressed on the instrument in
Hindu-Arabic numerals: "Over the whiche degrees ther ben noumbres of
augrim, that devyden thilke same degrees fro fyve to fyve," and "... the
nombres ... ben writen in augrim," meaning in the way of the algorism.
Thomas Usk about 1387 writes:[479] "a sypher in augrim have no might in
signification of it-selve, yet he yeveth power in signification to other."
So slow and so painful is the assimilation of new ideas.

Bernelinus[480] states that the abacus is a well-polished board (or table),
which is covered with blue sand and used by geometers in drawing
geometrical figures. We have previously mentioned the fact that the Hindus
also performed mathematical computations in the sand, although there is no
evidence to show that they had any column abacus.[481] For the purposes of
computation, Bernelinus continues, the board is divided into thirty
vertical columns, three of which are reserved for fractions. Beginning with
the units columns, each set of {122} three columns (_li