recommending his pupils to read Plato. He endeavoured to make logic generally comprehensible by freeing it of sophistical verbiage, and he very ingeniously made use of this new logic to inculcate in the minds of his pupils the maxims of the Reformation, for he was a Calvinist with fanatical tendencies. He was cited before the parliament, not for his religious opinions, but for his blasphemies against peripateticism, and though his trial

was not of an inquisitorial character, he was condemned, deprived of his professor's chair at the Royal College, and compelled to leave the country. His implacable enemies, Antonio de Govea, Jacques Charpentier, and others saw in him less the Huguenot than the detractor of Aristotle. Ramus, who had become the chief of the small school of Ramists, went to lecture in the

towns on the banks of the Rhine. After three years' exile he returned to France, and was included in the massacre of St. Bartholomew. His personal enemy, Jacques Charpentier, of Clermont (in the Oise), professor of mathematics at the Royal College, was accused of having had him massacred by his pupils during that terrible night.

Plato, notwithstanding the efforts of Ramus, had not many followers in the University of Paris, where scholasticism endeavoured to regain its sway. Aristotle continued to be the favourite of the school, and his philosophical

Fig. 54.-Battle of Beggars and Peasants over a Barrel of Wine in the Chapter headed, "Comment les vices se combattirent les uns aux autres pour les vivres."-Miniature of the "Roi Modus."-Manuscript of the Fifteenth Century.—Burgundy Library, Brussels.

predominance was fostered by the decrees of the Parliament and the royal ordinances. But the true French spirit was less in the direction of the study of logic, even reformed and renewed, than in moral philosophy, especially when it had a tendency to be sceptical and sarcastic (Fig. 54). Montaigne, at the close of the fifteenth century, was, so to speak, the founder of this philosophy, which neither denies nor affirms anything, which calls everything in question, and makes light of all subjects. He was born at the

Château de Montaigne, in the Périgord, upon the 28th of February, 1533. Though he attended all the classes at the College of Bordeaux, he may be said to have been self-taught, and to have become a philosopher in his own way through his intercourse with the poets, historians, and philosophers of antiquity. He delighted in the works of Seneca and Plutarch, but he would not "bite his nails over Aristotle, the monarch of modern doctrine." In after-years, when he wrote his immortal "Essays," he unhesitatingly declared against the dialectics of the schools—against every kind of doctrinal teaching. "It is pitiable," he writes, "that in our century philosophy should be, even for men of intelligence, a vain and fantastic name, which is without use or value in opinion or in fact. I believe that sophistry, by choking up the approaches to it, is the cause. It is a great mistake to depict it as inaccessible to children, of a forbidding countenance, full of frowns, and fearful to look at. Nothing can be more cheerful, sprightly, I was almost saying frolicsome." Michael de Montaigne inaugurated in France the philosophy of the libertines—that is to say, of the free-thinkers-different in some respects from that which François Rabelais professed, fifty years before, in his Pantagruelic works, and which John Calvin denounced as a pagan doctrine, accusing the libertines of atheism and impiety. "Scepticism," writes M. Hauréau, "had the last word in this propaganda in favour of the sprightly and almost frolicsome philosophy; and the young, only too easily led away by such remarks, gladly left, under the guidance of this new teacher, the arduous paths of study to revel in the intercourse of poets, and to turn the melancholy eyebrows of the logicians into derision."

MATHEMATICAL SCIENCES.

Ancient Systems of the Planetary World.-Ptolemy and Aristarchus of Samos.-Boethius, Pappus, and Gerbert.--Schools of Bagdad.—Mathematical School in Spain, Italy, England, and France.—Astronomical Researches of the Arabs.-Roger Bacon and Master Pierre.— Albertus Magnus and St. Thomas Aquinas.-Progress of Mathematics.-Popes and Kings protectors of the Exact Sciences.—The King of Hungary, Matthias Corvinus.—Principal Works composed in the Fifteenth Century.-Pic Mirandola.-Peter Ramus.-Tycho Brahe and Copernicus.

S a proof of the forward state of the exact sciences in the Middle Ages, it would be sufficient to instance a Roman basilica or a Gothic cathedral. What immensity and depth of mathematical calculations; what knowledge of geometry, statics, and optics; what experience and skill in execution must have been possessed by the architects and builders in hewing, carving, and fitting the stones, in raising them to great heights, in constructing enormous towers and gigantic belfries, in forming the many arches, some heavy and massive, others light and airy, in combining and neutralising the thrust of these arches which interlace and hide each other up to the very summit of the edifice-all as if the most complicated science had humbly made herself the servant of art, placing no obstacle in the way of its free development!

From the commencement of the Middle Ages and henceforward, mathematics were not so much the object of special and public teaching as of individual and solitary study, either in the shade of the cloisters or amidst associations of artisans who zealously preserved the traditions of their predecessors.

In the University centres, as in the Arab and Jewish schools which had

so much importance, practical science was generally made subordinate to speculative science. Thus the theory of the calculus, the formulæ of algebra, the projections of lines through space, the problems of triangulation, were by preference applied to astronomical observations, so that the transcendental mathematics were always inseparable from astronomy.

It was as follows that Claudius Ptolemæus, a Greek or Egyptian astronomer, constituted the mundane system in a "Cosmography" written in Greek, which became one of the bases of mathematical and astronomical science in the Middle Ages:-"The world is divided into two vast regions; the one ethereal, the other elementary. The ethereal region begins with the first mover, which accomplishes its journey from east to west in twenty-four hours; ten skies participate in this motion, and their totality comprises the double crystalline heaven, the firmament, and the seven planets." According to Ptolemæus, the double crystalline heaven was placed between the first mover and the firmament. The elementary region, comprising the four elements of fire, air, water, and earth, reigned beneath the cavity of the sky, and was subject to the influence of the moon. The terrestrial globe, composed of earth and water, existed motionless in the centre of the world, and was surrounded by the element of air, in which was mingled that of fire. This system was not, however, exclusively adopted by all the philosophers. Some of them accorded their preference to the system of Aristarchus of Samos, who did not place the earth in the centre of the world, and who attributed to it a rotary motion around the sun, which was suspended motionless amidst the planets and the planetary circles. According to Aristarchus of Samos, Mercury, the planet which is nearest the sun, completed his motion around it in three months, whilst Venus took seven months and a half to execute hers. The earth, apart from its motion round the sun in the space of a year, effected a second motion, revolving upon its own axis, in the space of twenty-four hours, thus causing the succession of day and night. The monthly motion of the moon around the earth was accomplished in about twenty-seven days. The fourth planet, Mars, took two years to accomplish his revolution round the sun; Jupiter, much farther distant, took twelve years, and Saturn thirty.

The system of Ptolemy eventually triumphed over that of Aristarchus, and at the close of the fifth century the great Boethius (Fig. 57), the favourite minister of Theodoric the Great, who loved and patronised literature and