THE HISTORY OF MATHEMATICS 389 



to have been made at least 2000 years B. C. They show a knowledge of 

 numbers which indicates that their civilization must have been far 

 removed from the low stages in which many native tribes exist at the 

 present day. Simple counting with the fingers of the two hands can 

 be considered as the first stage, but beyond ten some new system must 

 be devised. It appears that the Babylonians had learned the method 

 of position, that is, that the first figure to the right shall represent the 

 units, the next figure the tens and so on. They had even constructed 

 numbers with 60 as the base of the system instead of ten. They could 

 write numbers exceeding a million, one tablet giving a table of squares, 

 and they also used fractions. Their geometry, however, was only in 

 an elementary stage. But in astronomy they seem to have passed be- 

 yond the first stage of observation and to have been able to classify the 

 results, for they possessed a knowledge of the Saros or period of 

 18, 2/3 years in which the eclipses of the sun and moon recur; this 

 must have involved a long period of observation and record as well as 

 the ability to classify the results and it may perhaps be regarded as 

 the earliest recorded scientific deduction from observation. 



Concurrently with this civilization was one of perhaps equal de- 

 velopment in Egypt. The Ahmer Papyrus, which is usually dated 

 some 2000 years B. C, shows that the Egyptians had not only already 

 constructed an arithmetic but had started the solution of what we now 

 call equations of the first degree with one unknown. There was no 

 general method for solving the problems and no symbolism for the 

 unknown, although symbols for addition, subtraction and equality 

 occur. In geometry they were also somewhat in advance of the 

 Babylonians, apparently on account of practical needs in land survey- 

 ing. It is generally agreed that the pyramids show evidence of 

 astronomical observation in their orientation with respect to the stars 

 and they certainly show evidence of a knowledge of geometrical form. 

 As these monuments appear to go back some 4000 years B. C, we have 

 evidence of some considerable development much further back even 

 than the times indicated by the Babylonian tablets or the Ahmer 

 Papyrus. 



But the first real evidence of the scientific spirit comes from Greece. 

 While they probably inherited some of the accumulated knowledge of 

 both Babylonia and Egypt, they transformed much of the raw material 

 thus acquired into a finished product which has survived to our own 

 times. Much the most remarkable of all that we inherit from them 

 in science, is the change which they effected in the study of geometry. 

 The love and knowledge of form which is so strikingly exhibited in 

 their buildings and statuary was also applied to geometrical figures. 

 Development of logic and a real desire to know the sources of all 

 things, was applied to the same study. Thus Greek geometry was not 



