leadership of Raymond, then archbishop of Toledo. Among those who worked under his direction, John of Seville was most prominent. He translated works chiefly on Aristotelian philosophy. Of importance to us is a liber algorismi, compiled by him from Arabic authors. On comparing works like this with those of the abacists, we notice at once the most striking difference, which shows that the two parties drew from independent sources. It is argued by some that Gerbert got his apices and his arithmetical knowledge, not from Boethius, but from the Arabs in Spain, and that part or the whole of the geometry of Boethius is a forgery, dating from the time of Gerbert. If this were the case, then the writings of Gerbert would betray Arabic sources, as do those of John of Seville. But no points of resemblance are found. Gerbert could not have learned from the Arabs the use of the abacus, because all evidence we have goes to show that they did not employ it. Nor is it probable that he borrowed from the Arabs the apices, because they were never used in Europe except on the abacus. In illustrating an example in division, mathematicians of the tenth and eleventh centuries state an example in Roman numerals, then draw an abacus and insert in it the necessary numbers with the apices. Hence it seems probable that the abacus and apices were borrowed from the same source. The contrast between authors like John of Seville, drawing from Arabic works, and the abacists, consists in this, that, unlike the latter, the former mention the Hindoos, use the term algorism, calculate with the zero, and do not employ the abacus. The former teach the extraction of roots, the abacists do not; they teach the sexagesimal fractions used by the Arabs, while the abacists employ the duo-decimals of the Romans.[3]
A little later than John of Seville flourished Gerard of Cremona in Lombardy. Being desirous to gain possession of
