four books of Topical differences, two of Categorical syllogisms, one book of Divisions, and one of Definitions. And here the reader will not fail to observe that these logical lectures must have been the fruit of studies pursued not in Spain, but in France, for previous to Gerbert’s coming to Rheims, we have his own acknowledgment that he knew nothing of that science. After he had taken his scholars through this course, says Richer, he proceeded to initiate them into the art of rhetoric; and he set out on the principle, that in this branch of study a knowledge of the classical poets was essential. He therefore read and explained Virgil, Statius, and Terence; then the satirists, Juvenal, Persius, and Horace, and last of all, Lucan. After this, his pupils were exercised in disputation, which he taught with such art, that the art was never apparent; a thing, observes his biographer, which is held to be the perfection of oratory. Then he popularised the science of music;[141] and as to arithmetic, mathematics, and astronomy, he made these difficult studies easy and delightful. Richer devotes several pages to the description of the various instruments which he constructed, and by which he contrived to render the science of astronomy, as it were, sensible to the eyes of his scholars. A round wooden ball, _with its poles oblique to the horizon_, figured the world, the various astronomical and geographical phenomena being represented by other circles. In fact, from the minute description of the writer, we are obliged to conclude that Gerbert exhibited at his lectures two very passable specimens of the terrestrial and celestial globes. But the great boon, which he is commonly represented as bestowing on the European schools, was the introduction of that wonderful table, “in which nine ciphers represented all the numbers, and produced in their infinite combinations all multiplications and divisions.” This was the mystic _Abacus_, the foundation, no doubt, of our present system of numeration. It consisted of a tablet, on which three columns were marked out, sometimes in fixed lines, sometimes in sand sprinkled over its surface; and in these columns figures were arranged in units, tens and hundreds. The method in use for working out calculations, even with the assistance of this decimal system, as explained by Gerbert in several treatises, was, however, extremely intricate, though it was probably a vast improvement on the clumsy contrivances which had been resorted to by former scholars. How far, however, the Abacus is to be regarded as a new invention, appears more than doubtful. Its history has been made the subject of interesting modern researches, and the result seems to be that the system of numeration used and explained by Gerbert, contained nothing in it which had been unknown to Boëthius.[142] Nevertheless, he certainly seems to have elucidated and popularised the science of arithmetic, which from this epoch began to be more seriously studied.