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Gel'fand amazed me by talking of mathematics as though it were poetry. He once said about a long paper bristling with formulas that it contained the vague beginnings of an idea which he could only hint at and which he had never managed to bring out more clearly. I had always thought of mathematics as being much more straightforward: a formula is a formula, and an algebra is an algebra, but Gel'fand found hedgehogs lurking in the rows of his spectral sequences!

Mathematical Notices v. 38, no. 3, March 1991, pp. 185-7.

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# The Great Calculation According to the Indians, of Maximus Planudes

## Introduction

Maximus Planudes was born around 1255 in Nicomedia and died at Constantinople around 1305. He took the name Maximus, replacing his baptismal name of Manuel, when he became a monk, shortly before 1280. Apart from translating theological and classical works from Latin into Greek - a good knowledge of Latin seems to have been a rarity among the Byzantines -  he is best known for his editions and commentaries on Greek poetry and drama, as well as for his training of upcoming scholars, such as Manuel Moschopoulos, who continued the important work of preserving, and ensuring the survival of a number of important Greek works.

As with Moschopoulos, who wrote a work on Magic Squares, Planudes had an interest in Mathematics, evidenced by his editions of Aratos' PhainomenaTheodosios ' Sphairica, Euclid's Elements, (Ps-)Iamblichos' Theologoumena Arithmeticae and Diophantos ' Arithmetica.

The present work, The Great Calculation According to the Indians, introduces (i) the (eastern) Arabic form of the Indian numerals, as used in Persia, along with (ii) a detailed exposition of algorithms for addition, subtraction, multiplication and division, both in the decimal system of these numerals and also in the sexagesimal system (iii), whose applications lie in astronomy. Finally, he gives algorithms for the extraction of square roots (iv), to various degrees of accuracy.

The introduction of this numeral system to Europe cannot be traced down to any one person or event, but seems to have occured in various places independently and over a period of time. The earliest known European manuscript containing the first nine Hindu-Arabic numerals dates from 976 and was found in a monastery in northern Spain. Later, Abraham ben Meir ibn Ezra, after a residency in Toledo in the early twelfth century brought some form of the numeral system, (probably the western Arabic variety), to England sometime between 1140 and 1167, replacing the Arabic symbols with Hebrew letters, but maintaining the decimal structure.

Other figures such as Gerbert of Aurillac (c. 945-1003) and John of Hallifax  (Sacrobosco)(c. 1195-56) in France also played their part in the disemination of the new system.

Planudes may have acquired his knowledge of the numeral system and algorithms during his time in Venice, where he was stationed as Ambassador during the reign of the Byzantine Emperor Andronicos II. Venice was at that time, of course, a major trading city and a vital point of contact between the East and West.