Slonimski's Theorem

An observation due to Hayyim Selig Slonimski concerning the sequence of carry digits in a multiplication table. Namely, the sequence of carry digits is the Farey sequence.

This observation allowed Slonimski to create very compact multiplication tables for use in hand calculations. He received several awards for different devices for presenting these tables. The most common format were Joffe Bars similar to Napier's Rods. Joffe Bars were popular in Eastern Europe in the late 19th and early 20th century.

References

Weiss, Stephan (2011), "Slonimsky's Multiplying Device, an Impressive Example for Applied Mathematics" (PDF), Journal of the Oughtred Society, 20 (1): 23–30 . Provides a derivation of Slonimski's theorem, and some details on the calculating machine.
Knight, Henry (1847), Multiplication Tablets: Derived from a theorem of S. Slonimski (PDF), Birmingham: Josiah Allen and Son . Provides a complete set of tables.
Monnier, Valéry; Szrek, Walter; Zalewski, Janusz (2013), "Chaim Selig Slonimski and his adding devices", IEEE Ann. Hist. Comput., 35 (3): 42–53, MR 3111378

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