Page:Popular Science Monthly Volume 75.djvu/496

492 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 common 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, Romans, 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 children 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 decimal system. Separate symbols and words are found for $$60,\; 3{,}600$$ and $$\textstyle 21{,}600 \;(60,\; \overline{60^2},\; \overline{60^3})$$ and also for $$10,\; 600$$ and $$1{,}000$$ and $$36{,}000$$.

The ingenious hypothesis is advanced by M. Aures that the Babylonians having originally a decimal system, gradually changed from that system of numeration to the duo-decimal and then to the sexagesimal 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 Revolution, 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 systems 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 82, which is given as 1.4, 92 is given as