The preceding pages contain probably all the meagre facts from which it is still possible to discern how the Greeks came by their arithmetical nomenclature, both for whole numbers and for fractions. The subsequent progress of calculation, that is to say, the further use of the elementary processes, depends on many conditions which cannot well be satisfied without a neat and comprehensive visible symbolism. This boon the Greeks never possessed. Yet even without it a retentive memory and a clear logical faculty would suffice for the discovery of many important rules, such for instance as that, in a proportion, the product of the means is equal to the product of the extremes. It is probable, therefore, that much of the Greek arithmetical knowledge dates from a time far anterior to the works in which we find historical evidence of it. It is probable, again, that the Greeks derived from Egypt at an early date as many useful hints on arithmetic as they certainly did on geometry and other branches of learning. It becomes necessary, therefore, to introduce in this place some account of Egyptian arithmetic, both as showing at what date certain arithmetical rules were known to mankind and as providing a fund of knowledge from which the Greeks may have drawn very largely in prehistoric times. The facts to be now stated are in any case of great importance, since they furnish the only compact mass of evidence concerning the difficulties which beset ancient arithmetic and the way in which they were surmounted.

Quite recently a hieratic papyrus, included in the Rhind collection of the British Museum, has been deciphered and found to be a mathematical handbook, containing problems in arithmetic and geometry.