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 1 on arithmetic which are now in the National Library at Paris. He gave an account of the finger-symbolism 2 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

1 They have been edited by S. P. Tannery, Paris, 1886. 2 See above, page 113.

3 On the formation and history of magic squares, see my Mathematical Recreations, London, fourth edition, 1905, chap. v. On the work of Moschopulus, see S. Günther's Geschichte der mathematischen Wissenschaften, Leipzig, 1876, chap. iv.

integers be the consecutive numbers from 1 to n2, the square is said to be of the nth order, and in this case the sum of the numbers in any row, column, or diagonal is equal to an (n2 + 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.

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.

CHAPTER VII.

SYSTEMS OF NUMERATION AND PRIMITIVE ARITHMETIC.1

I HAVE in many places alluded to the Greek method of expressing numbers in writing, and I have thought it best to defer to this chapter the whole of what I wanted to say on the various systems of numerical notation which were displaced by the system introduced by the Arabs.

First, as to symbolism and language. The plan of indicating numbers by the digits of one or both hands is so natural that we find it in universal use among early races, and the members of all tribes now extant are able to indicate by signs numbers at least as high as ten: it is stated that in some languages the names for the first ten numbers are derived from the fingers used to denote them. For larger numbers we soon, however, reach a limit beyond which primitive man is unable to count, while as far as language goes it is well known that many tribes have no word for any number higher than ten, and some have no word for any number beyond four, all higher numbers being expressed by the words plenty or heap: in connection with this it is worth remarking that (as stated above) the Egyptians used the symbol for the word heap to denote an unknown quantity in algebra.

The number five is generally represented by the open hand,

1 The subject of this chapter has been discussed by Cantor and by Hankel. See also the Philosophy of Arithmetic by John Leslie, second edition, Edinburgh, 1820. Besides these authorities the article on Arithmetic by George Peacock in the Encyclopaedia Metropolitana, Pure Sciences, London, 1845; E. B. Tylor's Primitive Culture, London, 1873; Les signes numéraux et l'arithmétique chez les peuples de l'antiquité...by T. H. Martin, Rome, 1864; and Die Zahlzeichen...by G. Friedlein, Erlangen, 1869, should be consulted.

and it is said that in almost all languages the words five and hand are derived from the same root. It is possible that in early times men did not readily count beyond five, and things if more numerous were counted by multiples of it. Thus the Roman symbol X for ten probably represents two "V"s, placed apex to apex, and seems to point to a time when things were counted by fives.1 In connection with this it is worth noticing that both in Java and among the Aztecs a week consisted of five days.

The members of nearly all races of which we have now any knowledge seem, however, to have used the digits of both hands to represent numbers. They could thus count up to and including ten, and therefore were led to take ten as their radix of notation. In the English language, for example, all the words for numbers higher than ten are expressed on the decimal system: those for 11 and 12, which at first sight seem to be exceptions, being derived from Anglo-Saxon words for one and ten and two and ten respectively.

Some tribes seem to have gone further, and by making use of their toes were accustomed to count by multiples of twenty. The Aztecs, for example, are said to have done so. It may be noticed that we still count some things (for instance, sheep) by scores, the word score signifying a notch or scratch made on the completion of the twenty; while the French also talk of quatrevingts, as though at one time they counted things by multiples of twenty. I am not, however, sure whether the latter argument is worth anything, for I have an impression that I have seen the word octante in old French books; and there is no question 2 that septante and nonante were at one time common words for seventy and ninety, and indeed they are still retained in some dialects.

The only tribes of whom I have read who did not count in terms either of five or of some multiple of five are the Bolans of West Africa who are said to have counted by multiples of

1 See also the Odyssey, iv, 413-415, in which apparently reference is made to a similar custom.

2 See, for example, V. M. de Kempten's Practique...à ciffrer, Antwerp,