Modular arithmetic before C. F. Gauss: systematizations and discussions on remainder problems in 18th-century Germany
Remainder problems have a long tradition and were widely disseminated in books on calculation, algebra, and recreational mathematics. The earliest rules for solving come from ancient China, the general solution method for such problems is today called the Chinese remainder theorem. In Europe many singular solutions for particular cases are known. Bachet de Méziriac showed (1624) how these methods are connected with the Euclidian algorithm and with Diophantic equations. In the 18th-century Germany many scientists and university teacher dealt with the remainder problem, a long list of famous in their time mathematicians is given: Christlieb von Clausberg, C. Wolff, L. Euler, A. Kästner, W. Karsten, L. Lagrange, H. Olbers, C. F. Gauss, J. H. Lambert, C. F. Hindenburg. Clausberg systematized methods that used the greatest common divisor procedure. Euler and Lagrange used fraction solution methods. Hindenburg developed an new system based on finite, combinatorical principles and on direct production of the solution. He formulated the combinatory law of cyclic periods. This was a reach basis for Gauss to construct a general framework from 1791 onwards. The final version appeared in sections I and II of the Disquisitiones Arithmeticae (1801). The new formalism based on Gauss' methods of congruences. The paper is finished by for pages of useful references.
scientific article; zbMATH DE number 5532931
scientific article