INTRODUCTION 3 



written numeration, the Egyptians used the following 

 system. A special sign represented unity, another sign 

 represented ten, and so on. So that, if one had to 

 write the figure 23, it was necessary to repeat three 

 times the sign for unity and twice that for ten. 1 This 

 proceeding made writings singularly complicated. It 

 was the more inconvenient because the Egyptians had 

 not our abbreviated methods of multiplication and 

 division. For them multiplication was reduced to a 

 series of additions, and division to repeated subtrac- 

 tions. A further cause of complication in the calcula- 

 tions arose from the manner in which the fractions were 

 considered. The idea of a fraction must have been 

 evolved in the mind of man at a very early period. It 

 was imposed upon him as soon as he knew how to 

 measure a field, for it rarely happens that the unit 

 chosen as a measure is contained an exact number of 

 times in material objects. This being so, the idea of a 

 simple fraction can be conceived in two ways. One 

 may proceed as we do. In this case, the unit is under- 

 stood in the denominator which indicates the number 

 of subdivisions into which it is divided, while the 

 numerator shows the sum of the parts thus obtained 

 which one wishes to consider. To write t, for example, 

 is to say that of the seven subdivisions of the unit, one 

 only considers the sum of four of them. This being so, 

 to add or subtract two different fractions does not pre- 

 sent any difficulty. It is enough to reduce these 

 fractions to the same denominator, i.e. to the same 

 divisor of the unit, then to add or subtract the numera- 

 tors and the problem is solved. But it is possible, and 

 this is what the Egyptians did, to consider the fraction 

 as always representing a part of the same unit. In this 

 case the fractions will always have 1 for numerator, 

 the denominator indicating as before the number of 

 parts into which the unit is divided. Hence what we 

 1 22 Milhaud, Nouvelles Etudes, p. 51. 



