# Page:A History of Mathematics (1893).djvu/84

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THE GREEKS.

Greek writers seldom refer to calculation with alphabetic numerals. Addition, subtraction, and even multiplication were probably performed on the abacus. Expert mathematicians may have used the symbols. Thus Eutocius, a commentator of the sixth century after Christ, gives a great many multiplications of which the following is a specimen:

 ${\overline {\sigma \xi \epsilon }}$ ⁠ 2 6 5 ⁠ ${\overline {\sigma \xi \epsilon }}$ 2 6 5 δϺαϺ'β'α 40000, 12000, 1000 αϺ'β'γχτ 12000, 3600, 300 'ατκε 1000, 300, 25 ζϺσκε 70225

The operation is explained sufficiently by the modern numerals appended. In case of mixed numbers, the process was still more clumsy. Divisions are found in Theon of Alexandria's commentary on the Almagest. As might be expected, the process is long and tedious.

We have seen in geometry that the more advanced mathematicians frequently had occasion to extract the square root. Thus Archimedes in his Mensuration of the Circle gives a large number of square roots. He states, for instance, that ${{\sqrt {3}}<{1351 \over 730}}$ and ${{\sqrt {3}}>{265 \over 153}}$ , but he gives no clue to the method by which he obtained these approximations. It is not improbable that the earlier Greek mathematicians found the square root by trial only. Eutocius says that the method of extracting it was given by Heron, Pappus, Theon, and other commentators on the Almagest. Theon's is the only ancient method known to us. It is the same as the one used nowadays, except that sexagesimal fractions are employed in place of our decimals. What the mode of procedure actually was when sexagesimal fractions were not used, has been the subject of conjecture on the part of numerous modern writers.

Of interest, in connection with arithmetical symbolism, is the Sand-Counter (Arenarius), an essay addressed by Archi- 