BABYLONIA AND EGYPT 25 



prohibitive. Vestiges of the use of 5 and of 20 are familiar ; the 

 former, for example, in the Roman numerals IV, VI, etc., the latter 

 in such expressions as "three score and ten" and in the French 

 quatre-vingt. Increasing maturity of a tribe or race, as of an in- 

 dividual, is accompanied by gain in the command of larger and 

 larger numbers, the rate of progress being very dependent, however, 

 on a fortunate choice of notation. However great the capacity 

 for inventing number- words, it soon becomes necessary to employ 

 some system which shall lead to a regular development of higher 

 from lower names. The selection of a point at which dependent 

 names, and later, symbols shall begin, is one of the most important 

 steps in the history of mathematics. It is difficult for us to 

 realize the extent of our indebtedness to the comparatively recent 

 so-called Arabic or more properly, Hindu notation, in which 

 numbers of whatever magnitude may be expressed by means of 

 only ten symbols. In any case, however, the appreciation of 

 large numbers soon becomes vague. To most of us the word 

 million is nearly equivalent to an innumerable multitude. 



PRIMITIVE GEOMETRY. On the geometrical side data are 

 naturally more meagre. The notions of a primitive society in 

 regard to areas and perimeters and the ratio of a circumference 

 to its diameter may quite escape discovery. On the other hand 

 skill in making and reading maps is well known as among the 

 Esquimaux. 



RELATION OF GREEK TO OLDER CIVILIZATIONS. Mathematical 

 science seems to have first assumed definite form in Greece, and it 

 is of particular interest to study the indebtedness of the Greeks to 

 the older civilizations referred to in the preceding chapter. Some 

 degree of civilization doubtless existed further back than any records 

 run, in China, in India, in Babylonia, and in Egypt. But of these 

 only the latter two exerted a determining influence on the general 

 evolution of European science, India making minor though funda- 

 mental contributions at a much later stage. Babylonia and Egypt 

 exchanged ideas with each other, and, after unnumbered centuries, 

 furnished Greece with a certain nucleus of scientific knowledge of 

 which the Greeks made enormous use. In practical engineering 



