neae_ is the word
which Bernelinus uses) is grouped together by a semicircular arc placed
above them, while a smaller arc is placed over the units column and another
joins the tens and hundreds columns. Thus arose the designation _arcus
pictagore_[482] or sometimes simply _arcus_.[483] The operations of
addition, subtraction, and multiplication upon this form of the abacus
required little explanation, although they were rather extensively treated,
especially the multiplication of different orders of numbers. But the
operation of division was effected with some difficulty. For the
explanation of the method of division by the use of the complementary
difference,[484] long the stumbling-block in the way of the medieval
arithmetician, the reader is referred to works on the history of
mathematics[485] and to works relating particularly to the abacus.[486]

Among the writers on the subject may be mentioned Abbo[487] of Fleury (c.
970), Heriger[488] of Lobbes or Laubach {123} (c. 950-1007), and Hermannus
Contractus[489] (1013-1054), all of whom employed only the Roman numerals.
Similarly Adelhard of Bath (c. 1130), in his work _Regulae Abaci_,[490]
gives no reference to the new numerals, although it is certain that he knew
them. Other writers on the abacus who used some form of Hindu numerals were
Gerland[491] (first half of twelfth century) and Turchill[492] (c. 1200).
For the forms used at this period the reader is referred to the plate on
page 88.

After Gerbert's death, little by little the scholars of Europe came to know
the new figures, chiefly through the introduction of Arab learning. The
Dark Ages had passed, although arithmetic did not find another advocate as
prominent as Gerbert for two centuries. Speaking of this great revival,
Raoul Glaber[493] (985-c. 1046), a monk of the great Benedictine abbey of
Cluny, of the eleventh century, says: "It was as though the world had
arisen and tossed aside the worn-out garments of ancient time, and wished
to apparel itself in a white robe of churches." And with this activity in
religion came a corresponding interest in other lines. Algorisms began to
appear, and knowledge from the outside world found {124} interested
listeners. Another Raoul, or Radulph, to whom we have referred as Radulph
of Laon,[494] a teacher in the cloister school of his city, and the brother
of Anselm of Laon[495] the celebrated theologian, wrote a treatise on
music, extant but unpublished, and an arithme