denoted 24ths, 48ths and 36ths respectively. On the Greek abacus also the right-hand columns were of mixed systems which serve to make the calculating more difficult.
A late development of the same nature was the reckoning on lines which continued into the sixteenth century. The essentials are similar. A glance at the accompanying diagram explains the connection between this system and the decimal place system. The upper part represents the number 4,063, the lower part the number 3,251. It seems such a slight step, after acquiring special symbols for the groups of one
| M | C | X | I |
| 0 | 0 | 0 | |
| 0 | 0 | 0 | |
| 0 | 0 | 0 | |
| 0 | 0 | ||
| 0 | |||
| 0 | |||
| 0 | |||
| 0 | |||
| 0 | 0 | ||
| 0 | 0 | 0 | |
| 0 | 0 | 0 | 0 |
to nine to construct a symbol to indicate a blank space, but that step took centuries to achieve.
Of all ancient peoples the Hindus occupied themselves most deeply with numbers. To some of their scholars came the conception of a connection between the infinite of the universe and the infinite of numbers. This longing for the infinite found expression in the construction of ever increasing numbers. Buddha calculates the number of grains of sand in a mile and shows how to compute the number in a sphere whose radius is the distance to one of the fixed stars. Not content with this, the Buddha goes on to show how even greater numbers may be expressed, arriving at the equivalent in modern exponential notation of . The numerals of the ancient Tamils, who, like the mountaineers of Appalachian America, conserve the traditions of a more remote civilization, show us that the Hindu peoples originally had special symbols not only for the first nine units, but also for the nine tens, the nine hundreds and even the nine thousands. The formation of the sequences of large numbers revealed the futility of having separate signs for the mixed tens and hundreds, with the consequent result that they dropped the separate symbols for 20 to 90, 200 to 900, and used the pure decimal units in connection with the symbols for one to nine. A similar development took place in quite early times—pre-Christian—among the Chinese but their clumsy notation obscured the realization of the possibility of a simpler place system.
The reading of a large number in Hindu style reveals how close their nomenclature brought them to the place system.
