The Arabs were learned, but not original. Their chief service to science consists in this, that they adopted the learning of Greece and India, and kept what they received with scrupulous care. When the love for science began to grow in the Occident, they transmitted to the Europeans the valuable treasures of antiquity. Thus a Semitic race was, during the Dark Ages, the custodian of the Aryan intellectual possessions.

EUROPE DURING THE MIDDLE AGES.

With the third century after Christ begins an era of migration of nations in Europe. The powerful Goths quit their swamps and forests in the North and sweep onward in steady southwestern current, dislodging the Vandals, Sueves, and Burgundians, crossing the Roman territory, and stopping and recoiling only when reaching the shores of the Mediterranean. From the Ural Mountains wild hordes sweep down on the Danube. The Roman Empire falls to pieces, and the Dark Ages begin. But dark though they seem, they are the germinating season of the institutions and nations of modern Europe. The Teutonic element, partly pure, partly intermixed with the Celtic and Latin, produces that strong and luxuriant growth, the modern civilisation of Europe. Almost all the various nations of Europe belong to the Aryan stock. As the Greeks and the Hindoos-both Aryan races were the great thinkers of antiquity, so the nations north of the Alps became the great intellectual leaders of modern times.

Introduction of Roman Mathematics.

We shall now consider how these as yet barbaric nations of the North gradually came in possession of the intellectual

treasures of antiquity. With the spread of Christianity the Latin language was introduced not only in ecclesiastical but also in scientific and all important worldly transactions. Naturally the science of the Middle Ages was drawn largely from Latin sources. In fact, during the earlier of these ages Roman authors were the only ones read in the Occident. Though Greek was not wholly unknown, yet before the thirteenth century not a single Greek scientific work had been read or translated into Latin. Meagre indeed was the science which could be gotten from Roman writers, and we must wait several centuries before any substantial progress is made in mathematics.

After the time of Boethius and Cassiodorius mathematical activity in Italy died out. The first slender blossom of science among tribes that came from the North was an encyclopædia entitled Origines, written by Isidorus (died 636 as bishop of Seville). This work is modelled after the Roman encyclopædias of Martianus Capella of Carthage and of Cassiodorius. Part of it is devoted to the quadrivium, arithmetic, music, geometry, and astronomy. He gives definitions and grammatical explications of technical terms, but does not describe the modes of computation then in vogue. After Isidorus there follows a century of darkness which is at last dissipated by the appearance of Bede the Venerable (672-735), the most learned man of his time. He was a native of Ireland, then the home of learning in the Occident. His works contain treatises on the Computus, or the computation of Easter-time, and on finger-reckoning. It appears that a finger-symbolism was then widely used for calculation. The correct determination of the time of Easter was a problem which in those days greatly agitated the Church. It became desirable to have at least one monk at each monastery who could determine the day of religious festivals and could compute the calendar.

Such determinations required some knowledge of arithmetic. Hence we find that the art of calculating always found some little corner in the curriculum for the education of monks.

The year in which Bede died is also the year in which Alcuin (735-804) was born. Alcuin was educated in Ireland, and was called to the court of Charlemagne to direct the progress of education in the great Frankish Empire. Charlemagne was a great patron of learning and of learned men. In the great sees and monasteries he founded schools in which were taught the psalms, writing, singing, computation (computus), and grammar. By computus was here meant, probably, not merely the determination of Easter-time, but the art of computation in general. Exactly what modes of reckoning were then employed we have no means of knowing. It is not likely that Alcuin was familiar with the apices of Boethius or with the Roman method of reckoning on the abacus. He belongs to that long list of scholars who dragged the theory of numbers into theology. Thus the number of beings created by God, who created all things well, is 6, because 6 is a perfect number (the sum of its divisors being 1+2+3 = 6); 8, on the other hand, is an imperfect number (1 + 2 + 4 <8) ; hence the second origin of mankind emanated from the number 8, which is the number of souls said to have been in Noah's ark. There is a collection of "Problems for Quickening the Mind 99 (propositiones ad acuendos iuvenes), which are certainly as old as 1000 A.D. and possibly older. Cantor is of the opinion that they were written much earlier and by Alcuin. The following is a specimen of these "Problems": A dog chasing a rabbit, which has a start of 150 feet, jumps 9 feet every time the rabbit jumps 7. In order to determine in how many leaps the dog overtakes the rabbit, 150 is to be divided by 2. In this collection of problems, the areas of triangular and quadrangular pieces of land are found by the same formulas of

approximation as those used by the Egyptians and given by Boethius in his geometry. An old problem is the "cisternproblem" (given the time in which several pipes can fill a cistern singly, to find the time in which they fill it jointly), which has been found previously in Heron, in the Greek Anthology, and in Hindoo works. Many of the problems show that the collection was compiled chiefly from Roman sources. The problem which, on account of its uniqueness, gives the most positive testimony regarding the Roman origin is that on the interpretation of a will in a case where twins are born. The problem is identical with the Roman, except that different ratios are chosen. Of the exercises for recreation, we mention the one of the wolf, goat, and cabbage, to be rowed across a river in a boat holding only one besides the ferry-man. Query: How must he carry them across so that the goat shall not eat the cabbage, nor the wolf the goat? The solutions of the "problems for quickening the mind" require no further knowledge than the recollection of some few formulas used in surveying, the ability to solve linear equations and to perform the four fundamental operations with integers. Extraction of roots was nowhere demanded; fractions hardly ever occur.3

The great empire of Charlemagne tottered and fell almost immediately after his death. War and confusion ensued. Scientific pursuits were abandoned, not to be resumed until the close of the tenth century, when under Saxon rule in Germany and Capetian in France, more peaceful times began. The thick gloom of ignorance commenced to disappear. The zeal with which the study of mathematics was now taken up by the monks is due principally to the energy and influence of one man, Gerbert. He was born in Aurillac in Auvergne. After receiving a monastic education, he engaged in study, chiefly of mathematics, in Spain. On his return he taught

school at Rheims for ten years and became distinguished for his profound scholarship. By King Otto I. and his successors Gerbert was held in highest esteem. He was elected bishop of Rheims, then of Ravenna, and finally was made Pope under the name of Sylvester II. by his former pupil Emperor Otho III. He died in 1003, after a life intricately involved in many political and ecclesiastical quarrels. Such was the career of the greatest mathematician of the tenth century in Europe. By his contemporaries his mathematical knowledge was considered wonderful. Many even accused him of criminal intercourse with evil spirits.

Gerbert enlarged the stock of his knowledge by procuring copies of rare books. Thus in Mantua he found the geometry of Boethius. Though this is of small scientific value, yet it is of great importance in history. It was at that time the only book from which European scholars could learn the elements of geometry. Gerbert studied it with zeal, and is generally believed himself to be the author of a geometry. H. Weissenborn denies his authorship, and claims that the book in question consists of three parts which cannot come from one and the same author.21 This geometry contains nothing more than the one of Boethius, but the fact that occasional errors in the latter are herein corrected shows that the author had mastered the subject. "The first mathematical paper of the Middle Ages which deserves this name," says Hankel, “is a letter of Gerbert to Adalbold, bishop of Utrecht," in which is explained the reason why the area of a triangle, obtained "geometrically" by taking the product of the base by half its altitude, differs from the area calculated "arithmetically," according to the formula a (a + 1), used by surveyors, where a stands for a side of an equilateral triangle. He gives the correct explanation that in the latter formula all the small squares, in which the triangle is sup