492 THE POPULAR SCIENCE MONTHLY 



four as two and two, five as two and two and one, and six as two, two, 

 two. This system is found also among South American tribes. The 

 quinary system is the most frequent of all the systems occurring in the 

 numerals of American languages, although the twenty system is com- 

 mon along the Pacific. A study of the words of various American 

 Indian tribes reveals traces of a five system in the formation of the 

 words for six, seven and eight which are given as five and one, five and 

 two, and five and three. The higher numbers, however, are formed on 

 the decimal scale. The word for twenty signifies two tens and the 

 higher tens are similarly constructed. Among some of the African 

 tribes a partial five system is in use. Other tribes of northern Africa 

 have borrowed the decimal notation from their civilized neighbors. 



Without a single exception the ancient civilized peoples of all the 

 world — Egyptians, Babylonians, Hebrews, Chinese, Greeks, Eomans, 

 Hindus — all used the decimal systems. Such striking uniformity 

 among all the races of the earth requires a natural origin for the 

 decimal number base. As Herodotus first suggested, man counts by 

 tens because he has ten fingers. While there may be logical grounds 

 for the advocates of a duo-decimal system, the ten system is too deep- 

 rooted to be dislodged. Were we to acquire numbers as adults with 

 mature minds, a duo-decimal system might be possible, but with chil- 

 dren the acquisition of a twelve system may be said to be almost a 

 psychological impossibility. 



Among the Babylonians existed a sixty system mixed with a deci- 

 mal system. Separate symbols and words are found for 60, 3,600 and 

 21,600 (60, 60 2 , 60~ 3 ) and also for 10, 600 and 1,000 and 36,000. 



The ingenious hypothesis is advanced by M. Aures that the Baby- 

 lonians having originally a decimal system, gradually changed from 

 that system of numeration to the duo-decimal and then to the sexa- 

 gesimal in order to make the number system accord with their systems 

 of measurements. This is the reciprocal movement to that which is 

 taking place with us to-day and that which was effected for France by 

 the French Eevolution, the change from duo-decimal and what not else 

 systems of measurements to a decimal system in conformity with our 

 number system. The hypothesis of Aures is justified by the existence 

 of the special symbols and names for 10, 100 and 1,000, and many 

 other curious mixtures of decimal, duo-decimal and sexagesimal sys- 

 tems in the Babylonian measures. There is some comfort to be found 

 in the reflection that ours is not the first civilization to struggle with 

 diverse systems of notation and measurement. 



The most striking fact of Babylonian mathematics is that they were 

 in possession of a sixty place system. The famous tablets of Senkereh, 

 discovered by the English geologist, W. K. Loftus, give tables of 

 square and cubic numbers in cuneiform characters. In these tables the 

 numbers proceed regularly up to 8 2 , which is given as 1.4, 9 2 is given as 



