A Short Account of the History of Mathematics · W. W. Rouse Ball

Chapter 6: The Byzantine School

9 of 28 · 6 min of reading

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 pamphlet 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. He wrote a commentary on the first two books of the Arithmetic of Diophantus; a work on Hindoo arithmetic in which he used the Arabic numerals; 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. Barlaam seems to have been a man of great intelligence. He was sent as an ambassador to the Pope at Avignon, and acquitted himself creditably of a difficult mission; while there he taught Greek to Petrarch. He was famous at Constantinople for the ridicule he threw on the preposterous pretensions of the monks at Mount Athos who taught that those who joined them could, by steadily regarding their bodies, see a mystic light which was the essence of God. Barlaam advised them to substitute the light of reason for that of their bodies--a piece of advice which nearly cost him his life. The last of these monks was Isaac Argyrus, who died in 1372. He wrote three astronomical tracts, the manuscripts of which are in the libraries at the Vatican, Leyden, and Vienna: one on geodesy, the manuscript of which is at the Escurial: one on geometry, the manuscript of which is in the National Library at Paris: one on the arithmetic of Nicomachus, the manuscript of which is in the National Library at Paris: and one on trigonometry, the manuscript of which is in the Bodleian at Oxford.

Rhabdas. In the fourteenth or perhaps the fifteenth century Nicholas Rhabdas of Smyrna wrote two papers on arithmetic which are now in the National Library at Paris. He gave an account of the finger-symbolism which the Romans had introduced into the East and was then current there.

Pachymeres. Early in the fifteenth century Pachymeres wrote tracts on arithmetic, geometry, and four mechanical machines.

Moschopulus. A few years later Emmanuel Moschopulus, who died in Italy circ. 1460, wrote a treatise on magic squares. A magic square consists of a number of integers arranged in the form of a square so that the sum of the numbers in every row, in every column, and in each diagonal is the same. If the integers be the consecutive numbers from 1 to n^2 , the square is said to be of the n th order, and in this case the sum of the numbers in any row, column, or diagonal is equal to 1/2 n(n^2 + 1) . Thus the first 16 integers, arranged in either of the forms given below, form a magic square of the fourth order, the sum of the numbers in every row, every column, and each diagonal being 34.

1 & 15 & 14 & 4 12 & 6 & 7 & 9 8 & 10 & 11 & 5 13 & 3 & 2 & 16

15 & 10 & 3 & 6 4 & 5 & 16 & 9 14 & 11 & 2 & 7 1 & 8 & 13 & 12

In the mystical philosophy then current certain metaphysical ideas were often associated with particular numbers, and thus it was natural that such arrangements of numbers should attract attention and be deemed to possess magical properties. The theory of the formation of magic squares is elegant, and several distinguished mathematicians have written on it, but, though interesting, I need hardly say it is not useful. Moschopulus seems to have been the earliest European writer who attempted to deal with the mathematical theory, but his rules apply only to odd squares. The astrologers of the fifteenth and sixteenth centuries were much impressed by such arrangements. In particular the famous Cornelius Agrippa (1486--1535) constructed magic squares of the orders 3, 4, 5, 6, 7, 8, 9, which were associated respectively with the seven astrological "planets," namely, Saturn, Jupiter, Mars, the Sun, Venus, Mercury, and the Moon. He taught that a square of one cell, in which unity was inserted, represented the unity and eternity of God; while the fact that a square of the second order could not be constructed illustrated the imperfection of the four elements, air, earth, fire, and water; and later writers added that it was symbolic of original sin. A magic square engraved on a silver plate was often prescribed as a charm against the plague, and one (namely, that in the first diagram on the last page) is drawn in the picture of melancholy painted about the year 1500 by Albrecht Dürer. Such charms are still worn in the East.

Constantinople was captured by the Turks in 1453, and the last semblance of a Greek school of mathematics then disappeared. Numerous Greeks took refuge in Italy. In the West the memory of Greek science had vanished, and even the names of all but a few Greek writers were unknown; thus the books brought by these refugees came as a revelation to Europe, and, as we shall see later, gave a considerable stimulus to the study of science.

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End of part 9 of 28