M. A. Tsayger

This paper discusses the methods of writing numbers in ancient and pre-Petrine Russia and arithmetic operations in alphabetic numeral systems. The Russian numeral system, based on the Slavic Cyrillic alphabet, was modeled on the Byzantine one, where the Greek (Ionian) alphabetic-numeral system used letters of the Greek alphabet to write numbers. The essay compares the two related alphabetic-numerical systems and examines their differences. There is no known surviving abacus or similar counting board designed for arithmetic operations with alphabetic-numeral systems. The author offers a possible reconstruction of such a device and explains how it could perform addition, subtraction, and multiplication in the Slavic alphabetic system. The method is also compared to the multiplication of numbers in the Egyptian method of doubling and to the Greek multiplication table. The table uses so-called pythmenes, often ascribed to Pythagoras. The author believes that Pythagoras invented not the table itself, but a special procedure for converting multipliers into pythmenes, and the reverse procedure of converting pythmenes into the final result of the calculation. The resulting simplification of multiplication through what the author calls a «Pythagorean ladder» made alphabetic numbering especially attractive and promoted the distribution of the Greek alphabet and alphabetic numeral system to many regions in contact with ancient Greece and Byzantium, including the Near East, ancient Russia, Armenia, Georgia, and other places.