Mabch 18, 1910] 



SCIENCE 



431 



closely related work. This is the case, for ex- 

 ample, with four of the six cases from Harvard. 

 For the Oommittee on Collegiate Education 

 of Section L of the American Associa- 

 tion. 



Edward L. Thorndike, 



Chairman 

 Teachers College, 

 Columbia University 



THE SEXAGESIMAL SYSTEM AND THE 

 DIVISION OF THE CIRCLE 



The division of tlie hour and the degree into 

 60 equal parts, called minutes, and the minute 

 into 60 equal parts, called seconds, keeps fresh 

 in our minds the fact that the ancient Baby- 

 lonians used 60 as a base of numeration. Less 

 than ten years ago all seemed to agree on the 

 probable origin of this system. It was as- 

 sumed that the ancient Babylonians supposed 

 that there were only 360 days in a year and 

 hence divided the circle so that one day cor- 

 responded to each division. In support of this 

 hypothesis it was pointed out that the ancient 



Chinese divided the circle into 365i parts in 

 their Tcheou pei, and that this work could not 

 have been written before 213 B.C.; but at this 

 early date the Chinese were already ac- 

 quainted with the year of 365| days. From 

 the assumption that the circle was divided 

 into 360 equal parts before the origin of the 

 sexagesimal system, and the fact that the 

 radius of a circle can be applied exactly six 

 times as a chord of the circumference, it was 

 easy to account for the base 60. 



In recent years this question has received 

 considerable attention and many arguments 

 have been advanced against the given hypoth- 

 esis as regards the division of the circle. 

 These arguments appear convincing, but it is 

 not so easy to replace the old theory by one 

 which is free from objections. In the third 

 edition of his classic " Vorlesungen iiber 

 Geschichte der llathematik " (1907, volume 

 I., page 37) Moritz Cantor accepts the hypoth- 

 esis that the base 60 was selected as a conse- 

 quence of the mingling in the Babylonian 

 country of two ancient civilizations, one em- 

 ploying 10 and the other 6 as a base of numer- 

 ation. In view of the difficulties which this 

 hypothesis entails efforts have been made to 

 find a more plausible one. 



Professor Edmund H. Hoppe, Hamburg, 

 Germany, has recently advanced such a hypoth- 

 esis' and has given a large number of histor- 

 ical facts tending to its support. He assumes 

 that the normal angle among the ancient 

 Babylonians was an angle of an equilateral 

 triangle and that it was observed at an early 

 date that sis such angles cover the entire area 

 around a point. Hence the number 6 as- 

 sumed great importance, being regarded to 

 stand for completeness. The base 60 could 

 then have easily resulted from a division of 

 the normal angle into ten equal parts. After 

 this base was established, alongside the much 

 older base 10, the normal angle itself was di- 

 vided into 60 equal parts and this led to the 

 division of the circle into 360 equal parts. 



Whether this hypothesis will be generally 

 accepted remains to be seen. The fact that the 



' Archiv der Mathematik und Physik, Vol. 15 

 (1910), p. 304. 



