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 tendency to vigesimal forms, although, as a whole, the Germanic systems 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 examples of numbers which are wholly decimal, others wholly sexagesimal, 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 system 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 render 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 explain 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 acquainted 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 Archæology for May, 1883, pages 120, 121, says, in substance: 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-
