THE ORIGIN OF NUMBER SYSTEMS. 533 



to be no other good reason why 20 should have been adopted 

 for a base. The most perfect examples of vigesimal scales are 

 those of the Mayas of Yucatan and of the Aztecs of Mexico. It 

 has already been mentioned that traces of this system are to be 

 found in our own English numerals and in those of the French. 

 Danish and some of the kindred languages show a strong tend- 

 ency to vigesimal forms, although, as a whole, the Germanic sys- 

 tems of counting are purely decimal. 



Among the important number systems of the world there is 

 one which uses neither 5, 10, nor 20 as its base namely, the 

 sexagesimal scale of the ancient Babylonians. This system is 

 of special interest to ourselves, for its influence is still felt in 

 the division of our degree into 60 minutes, and the minute into 

 60 seconds. It seems to have arisen and continued in use side 

 by side with a decimal system, for the monuments furnish ex- 

 amples of numbers which are wholly decimal, others wholly sexa- 

 gesimal, and still others in which the two systems are combined. 

 It is a question of great interest to know how such a system came 

 to be adopted. It seems reasonable to suppose that it was formed 

 artificially that is, 60 did not come to be the base of this sys- 

 tem by a process of natural development, as 5, 10, or 20 came 

 to be the bases in the systems of other races. In all probability, 

 therefore, it grew up after the decimal system, and may have 

 been invented for the purposes of astronomical calculation, for 

 the Babylonians were famous astronomers in their day. It is 

 not impossible to suppose that its purpose originally was to ren- 

 der the calculations of the astronomers less intelligible to those 

 who were acquainted with only the decimal scale. However that 

 may have been, its use apparently became common. M. Cantor, 

 the German writer on the History of Mathematics, seeks to ex- 

 plain its origin by saying that the Babylonians divided the circle 

 of the heavens into 360 degrees, one degree for each of the 360 days 

 into which they divided the year. They were probably also ac- 

 quainted with the fact that the chord equal to the radius subtends 

 exactly one sixth of the circumference, or 60 degrees. This may 

 have led to the adoption of 60 as the number base. Prof. John P. 

 Peters, in a letter published in the Proceedings of the Society of 

 Biblical Archaeology for May, 1883, pages 120, 121, says, in sub- 

 stance : The use of the fingers of one hand to count to 5 was in 

 some cases extended to 6, by using the open hand with the fingers 

 and thumb extended to express 5, and then indicating 6 by the 

 closed hand. This method, if extended to both hands, gives 

 rise ordinarily to a duodecimal system ; and we have abundant 

 evidence both in our own English and in some other languages of 

 the presence of a duodecimal element, which may have arisen in 

 the way suggested. The Babylonians, however, instead of devel- 



