Checks on Computations

As we have mentioned, the creator of congruence theory was the German mathematician Gauss. His famous work on number theory, the Disquisitiones Arithmeticae, appeared in 1801 when he was twentyfour years old. The Disquisitiones has recently been translated into English (Yale University Press), so if you are interested in reading parts of one of the masterpieces of mathematics you may now do so. The first chapters deal with congruence theory and you have already learned enough about congruences to be able to follow Gauss's presentation.

But let us not fail to mention that there are traces of congruence theory centuries before the time of Gauss. Some of these appear in the ancient check rules for arithmetical computations. They formed an integral part of the instruction in arithmetic in the Renaissance. Some of them are still in use, and for all we know about their origin they may have their roots in antiquity.

How they originally were introduced we don't know, but let us indicate a plausible way in which they may have been discovered. We go back to the times of the computing boards. On such an abacus each digit in the numbers needed for the calculations would be laid out by means of counters or stones or sticks or nuts, each group marking the number of units, tens, hundreds, and so on, according to its place.