the labors of Hercules or of Socrates?" Books came to be scarce. But the decline of education was not universal. If studies failed in Gaul or Italy, they flourished in Ireland and afterward in Britain, and returned later from these outer borders to the old central lands of the Empire. Further, in spite of depression and discouragement, there was a continuity of learning even in the darkest ages and countries. Certain school books hold their ground. . . Capella. . . Boethius. Cassiodorus. . . And later Isodorus of Seville with a number of other authors are found in the ages of distress and anarchy more or less calmly giving their lectures and preserving the standards of a liberal education. Much of this work was humble enough, but it was of great importance for the times that came after. The darkest ages, with all their negligence, kept alive the life of the ancient world. Boethius [in the sixth century A.D., see p. 148] is the interpreter of the ancient world and its wisdom, accepted by all the tribes of Europe from one age to another, and never disqualified in his office of teacher even by the most subtle and elaborate theories of the later schools. . . . Cassiodorus (490-585) is wanting in the graces of Boethius, and he is much sooner forgotten; but his enormous industry, his organization of literary production, his educational zeal have all left their effects indelibly in modern civilization. By his definition of the seven Liberal Arts, and by his examples of methods in teaching them, he is the spiritual author of the universities, the patron of all the available learning in the world. — Ker, Dark Ages. THE ESTABLISHMENT OF SCHOOLS BY CHARLEMAGNE. - We have seen above how the schools of Athens were closed by Justinian in 529. Such schools as existed after that time were chiefly ecclesiastical and their teachings opposed to pagan or heathen (i.e. Greek) learning. At length, however, in 787 Charlemagne, moved it is said by the troublesome variety of writing as well as the general illiteracy of his people, ordered the establishment of schools in connection with every abbey of his realm, and summoned to take charge of them Peter of Pisa and Alcuin of York (735-804) (called by Guizot "the intellectual prime minister of Charlemagne"), whose names stand among the highest in a revival of learning thus begun in western Europe. In the later part of the eighth century begins the great age of medieval learning, the educational work of Charles the Great. There was some leisure and freedom and much literary ambition. The Latin poets of the court of Charlemagne have an enthusiasm and delight in classical poetry. . . . In prose there was no less activity. Besides the scientific treatises and the commentaries, the edifying works of Alcuin and others, there were histories. . . . The scholarly spirit of the ninth century. is not limited to the orthodox routine. One of the chief scholars, with more Greek than most others, Erigena, is famous for more than his learning, as a philosopher, who, whatever his respect for the Church, acknowledged no authority higher than reason. - Ker. Alcuin himself taught rhetoric, logic, mathematics and divinity, becoming master of the great school at St. Martin's of Tours. Of his arithmetic the following problem is an illustration: If 100 bushels of corn are distributed among 100 people in such a manner that each man receives 3 bushels, each woman 2, and each child half a bushel; how many men, women and children are there? Of six possible solutions Alcuin gives but one. The mathematics taught in Charlemagne's schools would naturally include the use of the abacus, the multiplication table, and the geometry of Boethius. Beyond this, a little Latin with reading and writing sufficed for the needs of the church and her servants, and was supplemented by music and theology for her higher officers. The recognized intellectual needs of the world were indeed but slight. The civilization of Rome had been gradually submerged by successive waves of barbaric invasion from the north, as a similar fate was soon to be met by the still higher culture of Alexandria. The best intellect of the times was perforce drawn into other forms of activity, while such scholars as remained found no favorable environment for fruitful study. The Benedictine monasteries, indeed, sheltered a few studious monks whose scientific interest scarcely extended beyond the mathematics necessary for their simple accounts, and the computation connected with the determination of the date of Easter. Near the close of the tenth century Gerbert of Aquitaine (940– 1003), afterwards Pope Sylvester II, devoted his versatile genius in part to mathematical science. He constructed not only abaci, but terrestrial and celestial globes, and collected a valuable library. To him were also attributed a clock, and an organ worked by steam. He wrote works on the use of the abacus, on the division of numbers and on geometry. The last named contains a solution of the relatively difficult problem to find the sides of a right triangle whose hypotenuse and area are given. Unfortunately the latter part of his life was absorbed in political intrigue and his death in 1003 cut short his plans for attempting the recovery of the Holy Land. Out of the schools of Charlemagne gradually grew up that subtle, minute and over-refined learning of the later Middle Ages which has come to be known as Scholasticism. Based as it was upon authority instead of experiment, and magnifying, as it did, details more than principles, it sharpened rather than broadened the intellect, and was indifferent if not unfavorable to science. REFERENCES FOR READING LUCRETIUS. On the Nature of Things. STRABO. Geography. PLINY. Natural History. FRONTINUS. The Waterworks of Rome. (Tr. by C. Herschel.) CHAPTER VIII HINDU AND ARABIAN SCIENCE. THE MOORS IN SPAIN The grandest achievement of the Hindus and the one which, of all mathematical investigations, has contributed most to the general progress of intelligence, is the invention of the principle of position in writing numbers. - Cajori. Indeed, if one understands by algebra the application of arithmetical operations to composite magnitudes of all kinds, whether they be rational or irrational number or space magnitudes, then the learned Brahmins of Hindustan are the true inventors of algebra. Hankel. In the ninth century the School of Bagdad began to flourish, just when the Schools of Christendom were falling into decay in the West and into decrepitude in the East. The newly-awakened Moslem intellect busied itself at first chiefly with Mathematics and Medical Science; afterwards Aristotle threw his spell upon it, and an immense system of orientalized Aristotelianism was the result. From the East, Moslem learning was carried to Spain; and from Spain Aristotle reëntered Northern Europe once more, and revolutionized the intellectual life of Christendom far more completely than he had revolutionized the intellectual life of Islam. Rashdall. ALEXANDRIA fell to the Arabs in 641 A.D." As a matter of historical perspective it is noteworthy that the interval between its foundation by Alexander the Great and its capture by the Mohammedans, - during most of which period it was the intellectual centre of the world, — is almost equal to that between Charlemagne's time and our own. The preservation and transmission of portions of Greek science through the Dark Ages to the dawn of science in western Europe about 1200 A.D. was mainly effected through three distinct, though not quite independent, channels. First, there was to a limited extent a direct inheritance of ancient learning within the Italian peninsula, through all its political and military turmoil. Second, a substantial legacy was received indirectly through the Moors in Spain; while, third, additions of great importance came later through Italy from Constantinople. Before following the direct Latin-Italian line a brief sketch of Hindu and Arabic science is desirable. HINDU MATHEMATICS. The far-reaching conquests of Alexander the Great (330 B.C.) immensely stimulated communication of ideas between the Mediterranean world and Asia, and the East was able to make certain great contributions to mathematical science just where the Greeks were relatively weakest, namely in arithmetic and the rudiments of algebra and trigonometry. Several centuries before our era the Pythagorean theorem and an excellent approximation for √2 were known in India in connection with the rules for the construction of altars. The mathematicians however from whom we trace the later development of mathematics date from the sixth and following centuries. About 530 A.D. Arya-bhata wrote a book in four parts dealing with astronomy and the elements of spherical trigonometry, and enunciating numerous rules of arithmetic, algebra and plane trigonometry. He gives the sums of the series - solves quadratic equations, gives a table of sines of successive multiples of 33° - i.e. twenty-fourths of a right angle, and even uses the value = 3.1416, correct to five places. His geometry is in general inferior. Some years later, Brahmagupta composed a system of astronomy in verse, with two chapters on mathematics. In this he discusses arithmetical progression, quadratic equations, areas of triangles, quadrilaterals and circles, volume and surface of pyramids and cones. His value of π is √10 = 3.16+. Typical problems and discussions are the following: Two apes lived at the top of a cliff of height 100, whose base was distant 200 from a neighboring village. One descended the cliff, and walked to the village, the other flew up a height x and then flew in a |