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there were thus suffered to continue, though their existence was of a precarious character. Under these conditions mathematics

continued to be read in Egypt for another hundred years, but all interest in the study had gone.

Roman Mathematics.1

I ought not to conclude this part of the history without any mention of Roman mathematics, for it was through Rome that mathematics first passed into the curriculum of medieval Europe, and in Rome all modern history has its origin. There is, however, very little to say on the subject. The chief study of the place was in fact the art of government, whether by law, by persuasion, or by those material means on which all government ultimately rests. There were, no doubt, professors who could teach the results of Greek science, but there was no demand for a school of mathematics. Italians who wished to learn more than the elements of the science went to Alexandria or to places which drew their inspiration from Alexandria.

The subject as taught in the mathematical schools at Rome seems to have been confined in arithmetic to the art of calculation (no doubt by the aid of the abacus) and perhaps some of the easier parts of the work of Nicomachus, and in geometry to a few practical rules; though some of the arts founded on a knowledge of mathematics (especially that of surveying) were carried to a high pitch of excellence. It would seem also that special attention was paid to the representation of numbers by signs. The manner of indicating numbers up to ten by the use of fingers must have been in practice from quite early times, but about the first century it had been developed by the Romans into a finger-symbolism by which numbers up to 10,000 or perhaps more could be represented: this would seem to have been taught in the Roman schools. It is described by Bede, and therefore would seem to have been known as far west as

1 The subject is discussed by Cantor, chaps. xxv, xxvi, and xxvii; also by Hankel, pp. 294-304.

Britain; Jerome also alludes to it; its use has still survived in the Persian bazaars.

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I am not acquainted with any Latin work on the principles of mechanics, but there were numerous books on the practical side of the subject which dealt elaborately with architectural and engineering problems. We may judge what they were like by the Mathematici Veteres, which is a collection of various short treatises on catapults, engines of war, &c. and by the KeoToí, written by Sextus Julius Africanus about the end of the second century, part of which is included in the Mathematici Veteres, which contains, amongst other things, rules for finding the breadth of a river when the opposite bank is occupied by an enemy, how to signal with a semaphore, &c.

In the sixth century Boethius published a geometry containing a few propositions from Euclid and an arithmetic founded on that of Nicomachus; and about the same time Cassiodorus discussed the foundation of a liberal education which, after the preliminary trivium of grammar, logic, and rhetoric, meant the quadrivium of arithmetic, geometry, music, and astronomy. These works were written at Rome in the closing years of the Athenian and Alexandrian schools, and I therefore mention them here, but as their only value lies in the fact that they became recognized text-books in medieval education I postpone their consideration to chapter VIII.

Theoretical mathematics was in fact an exotic study at Rome; not only was the genius of the people essentially practical, but, alike during the building of their empire, while it lasted, and under the Goths, all the conditions were unfavourable to abstract science.

On the other hand, Alexandria was exceptionally well placed to be a centre of science. From the foundation of the city to its capture by the Mohammedans it was disturbed neither by foreign nor by civil war, save only for a few years when the rule of the Ptolemies gave way to that of Rome: it was wealthy, and its rulers took a pride in endowing the university and lastly, just as in commerce it became the meeting-place of the east and the west, so it had the good fortune to be the dwelling

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place alike of Greeks and of various Semitic people; the one race shewed a peculiar aptitude for geometry, the other for sciences which rest on measurement. Here too, however, as time went on the conditions gradually became more unfavourable, the endless discussions on theological dogmas and the increasing insecurity of the empire tending to divert men's thoughts into other channels.

End of the Second Alexandrian School.

The precarious existence and unfruitful history of the last two centuries of the second Alexandrian School need no record. In 632 Mohammed died, and within ten years his successors had subdued Syria, Palestine, Mesopotamia, Persia, and Egypt. The precise date on which Alexandria fell is doubtful, but the most reliable Arab historians give December 10, 641— a date which at any rate is correct within eighteen months.

With the fall of Alexandria the long history of Greek mathematics came to a conclusion. It seems probable that the greater part of the famous university library and museum had been destroyed by the Christians a hundred or two hundred years previously, and what remained was unvalued and neglected. Some two or three years after the first capture of Alexandria a serious revolt occurred in Egypt, which was ultimately put down with great severity. I see no reason to doubt the truth of the account that after the capture of the city the Mohammedans destroyed such university buildings and collections as were still left. It is said that, when the Arab commander ordered the library to be burnt, the Greeks made such energetic protests that he consented to refer the matter to the caliph Omar. The caliph returned the answer, "As to the books you have mentioned, if they contain what is agreeable with the book of God, the book of God is sufficient without them; and, if they contain what is contrary to the book of God, there is no need for them; so give orders for their destruction." The account goes on to say that they were burnt in the public baths of the city, and that it took six months to consume them all.

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CHAPTER VI.

THE BYZANTINE SCHOOL. 641-1453.

It will be convenient to consider the Byzantine school in connection with the history of Greek mathematics. After the capture of Alexandria by the Mohammedans the majority of the philosophers, who previously had been teaching there, migrated to Constantinople, which then became the centre of Greek learning in the East and remained so for 800 years. But though the history of the Byzantine school stretches over so many years a period about as long as that from the Norman Conquest to the present day-it is utterly barren of any scientific interest; and its chief merit is that it preserved for us the works of the different Greek schools. The revelation of these works to the West in the fifteenth century was one of the most important sources of the stream of modern European thought, and the history of the Byzantine school may be summed up by saying that it played the part of a conduit-pipe in conveying to us the results of an earlier and brighter age.

The time was one of constant war, and men's minds during the short intervals of peace were mainly occupied with theological subtleties and pedantic scholarship. I should not have mentioned any of the following writers had they lived in the Alexandrian period, but in default of any others they may be noticed as illustrating the character of the school. I ought also,

perhaps, to call the attention of the reader explicitly to the fact that I am here departing from chronological order, and that the mathematicians mentioned in this chapter were contemporaries of those discussed in the chapters devoted to the mathematics of the middle ages. The Byzantine school was so isolated that I deem this the best arrangement of the subject.

Hero. One of the earliest members of the Byzantine school was Hero of Constantinople, circ. 900, sometimes called the younger to distinguish him from Hero of Alexandria. Hero would seem to have written on geodesy and mechanics as applied to engines of war.

During the tenth century two emperors, Leo VI. and Constantine VII., shewed considerable interest in astronomy and mathematics, but the stimulus thus given to the study of these subjects was only temporary.

Psellus. In the eleventh century Michael Psellus, born in 1020, wrote a pamphlet1 on the quadrivium: it is now in the National Library at Paris.

In the fourteenth century we find the names of three monks who paid attention to mathematics.

Planudes. Barlaam. Argyrus. The first of the three was Maximus Planudes.2 He wrote a commentary on the first two books of the Arithmetic of Diophantus; a work on Hindoo arithmetic in which he introduced the use of the Arabic numerals into the Eastern empire; and another on proportions which is now in the National Library at Paris. The next was a Calabrian monk named Barlaam, who was born in 1290 and died in 1348. He was the author of a work, Logistic, on the Greek methods of calculation from which we derive a good deal of information as to the way in which the Greeks treated numerical fractions.3 Barlaam seems to have been a man of great 1 It was printed at Bâle in 1556. Psellus also wrote a Compendium Mathematicum which was printed at Leyden in 1647.

2 His arithmetical commentary was published by Xylander, Bâle, 1575: his work on Hindoo arithmetic, edited by C. J. Gerhardt, was published at Halle, 1865.

3 Barlaam's Logistic, edited by Dasypodius, was published at Strassburg, 1572; another edition was issued at Paris in 1600.

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