and subtractions. Thus, as late as 944, a certain mathematician who in the course of his work wants to multiply 400 by 5 finds the result by addition. The same writer, when he wants to divide 6152 by 15, tries all the multiples of 15 until he gets to 6000, this gives him 400 and a remainder 152; he then begins again with all the multiples of 15 until he gets to 150, and this gives him 10 and a remainder 2. Hence the answer is 410 with a remainder 2.

A few mathematicians, however, such as Hero of Alexandria, Theon, and Eutocius, multiplied and divided in what is essentially the same way as we do. Thus to multiply 18 by 13 they proceeded as follows:

I suspect that the last step, in which they had to add four numbers together, was obtained by the aid of the abacus.

These, however, were men of exceptional genius, and we must recollect that for all ordinary purposes the art of calculation was performed only by the use of the abacus and the multiplication table, while the term arithmetic was confined to the theories of ratio, proportion, and of numbers.

All the systems here described were more or less clumsy, and they have been displaced among civilized races by the Arabic system in which there are ten digits or symbols, namely, nine for the first nine numbers and another for zero. In this system an integral number is denoted by a succession of digits, each digit representing the product of that digit and a power of ten, and the number being equal to the sum of these products. Thus, by means of the local value attached to nine symbols and a symbol for zero, any number in the decimal scale of notation can be expressed. The history of the development of the science of arithmetic with this notation will be considered below in chapter XI.

SECOND PERIOD.

Mathematics of the Middle Ages and Renaissance.

This period begins about the sixth century, and may be said to end with the invention of analytical geometry and of the infinitesimal calculus. The characteristic feature of this period is the creation or development of modern arithmetic, algebra, and trigonometry.

In this period I consider first, in chapter VIII, the rise of learning in Western Europe, and the mathematics of the middle ages. Next, in chapter IX, I discuss the nature and history of Hindoo and Arabian mathematics, and in chapter x their introduction into Europe. Then, in chapter XI, I trace the subsequent progress of arithmetic to the year 1637. Next, in chapter XII, I treat of the general history of mathematics during the renaissance, from the invention of printing to the beginning of the seventeenth century, say, from 1450 to 1637; this contains an account of the commencement of the modern treatment of arithmetic, algebra, and trigonometry. Lastly, in chapter XIII, I consider the revival of interest in mechanics, experimental methods, and pure geometry which marks the last few years of this period, and serves as a connecting link between the mathematics of the renaissance and the mathematics of modern times.

K

CHAPTER VIII.

THE RISE OF LEARNING IN WESTERN EUROPE.1 1 CIRC. 600-1200.

Education in the sixth, seventh, and eighth centuries.

THE first few centuries of this second period of our history are singularly barren of interest; and indeed it would be strange if we found science or mathematics studied by those who lived in a condition of perpetual war. Broadly speaking we may say that from the sixth to the eighth centuries the only places of study in western Europe were the Benedictine monasteries. We may find there some slight attempts at a study of literature; but the science usually taught was confined to the use of the abacus, the method of keeping accounts, and a knowledge of the rule by which the date of Easter could be determined. Nor was this unreasonable, for the monk had renounced the world, and there was no reason why he should learn more science than was required for the services of the Church and his monastery. The traditions of Greek and Alexandrian learning gradually died away. Possibly in Rome and a few favoured places copies of the works of the great Greek mathematicians were obtain

1 The mathematics of this period has been discussed by Cantor, by S. Günther, Geschichte des mathematischen Unterrichtes im deutschen Mittelalter, Berlin, 1887; and by H. Weissenborn, Gerbert, Beiträge zur Kenntniss der Mathematik des Mittelalters, Berlin, 1888.

able, though with difficulty, but there were no students, the books were unvalued, and in time became very scarce.

Three authors of the sixth century-Boethius, Cassiodorus, and Isidorus-may be named whose writings serve as a connecting link between the mathematics of classical and of medieval times. As their works remained standard text-books for some six or seven centuries it is necessary to mention them, but it should be understood that this is the only reason for doing so; they show no special mathematical ability. It will be noticed that these authors were contemporaries of the later Athenian and Alexandrian schools.

Boethius. Anicius Manlius Severinus Boethius, or as the name is sometimes written Boetius, born at Rome about 475 and died in 526, belonged to a family which for the two preceding centuries had been esteemed one of the most illustrious in Rome. It was formerly believed that he was educated at Athens: this is somewhat doubtful, but at any rate he was exceptionally well read in Greek literature and science.

He

Boethius would seem to have wished to devote his life to literary pursuits; but recognizing "that the world would be happy only when kings became philosophers or philosophers kings," he yielded to the pressure put on him and took an active share in politics. He was celebrated for his extensive charities, and, what in those days was very rare, the care that he took to see that the recipients were worthy of them. was elected consul at an unusually early age, and took advantage of his position to reform the coinage and to introduce the public use of sun-dials, water-clocks, etc. He reached the height of his prosperity in 522 when his two sons were inaugurated as consuls. His integrity and attempts to protect the provincials from the plunder of the public officials brought on him the hatred of the Court. He was sentenced to death while absent from Rome, seized at Ticinum, and in the baptistery of the church there tortured by drawing a cord round his head till the eyes were forced out of the sockets, and finally beaten to death with clubs on October 23, 526. Such at least is the