Among the writers on the subject may be mentioned Abbo[[487]] of Fleury (c. 970), Heriger[[488]] of Lobbes or Laubach
(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
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of his city, and the brother of Anselm of Laon[[495]] the celebrated theologian, wrote a treatise on music, extant but unpublished, and an arithmetic which Nagl first published in 1890.[[496]] The latter work, preserved to us in a parchment manuscript of seventy-seven leaves, contains a curious mixture of Roman and ġobār numerals, the former for expressing large results, the latter for practical calculation. These ġobār "caracteres" include the sipos (zero),
At the same time the words algorismus and cifra were coming into general use even in non-mathematical literature. Jordan [[497]] cites numerous instances of such use from the works of Alanus ab Insulis[[498]] (Alain de Lille), Gautier de Coincy (1177-1236), and others.
Another contributor to arithmetic during this interesting period was a prominent Spanish Jew called variously John of Luna, John of Seville, Johannes Hispalensis, Johannes Toletanus, and Johannes Hispanensis de Luna.[[499]]
His date is rather closely fixed by the fact that he dedicated a work to Raimund who was archbishop of Toledo between 1130 and 1150.[[500]] His interests were chiefly in the translation of Arabic works, especially such as bore upon the Aristotelian philosophy. From the standpoint of arithmetic, however, the chief interest centers about a manuscript entitled Joannis Hispalensis liber Algorismi de Practica Arismetrice which Boncompagni found in what is now the Bibliothèque nationale at Paris. Although this distinctly lays claim to being Al-Khowārazmī's work,[[501]] the evidence is altogether against the statement,[[502]] but the book is quite as valuable, since it represents the knowledge of the time in which it was written. It relates to the operations with integers and sexagesimal fractions, including roots, and contains no applications.[[503]]
Contemporary with John of Luna, and also living in Toledo, was Gherard of Cremona,[[504]] who has sometimes been identified, but erroneously, with Gernardus,[[505]] the
author of a work on algorism. He was a physician, an astronomer, and a mathematician, translating from the Arabic both in Italy and in Spain. In arithmetic he was influential in spreading the ideas of algorism.