*THE POPULAR SCIENCE MONTHLY*

confusion, denoted ten by drawing a tenth stroke over the nine parallel ones,

and that this was abbreviated to two strokes crossed, X. The upholders of this theory assert that five, half of ten, was then denoted by V, half of X. For fifty, L was used; for one hundred, C; for five hundred, D; for one thousand, M. The numbers in between were expressed by combination: LX = 60, DCXV = 615; or by addition to or subtraction from the nearest one of the seven symbols: XII = 12, IX = 9.

The advantage of this system lay in the fewness of the symbols employed. Where the Hebrew had twenty-two characters and the Greek twenty-seven, the Roman made use only of seven. Because of this fewness the value of the characters could be easily remembered. On the other hand the smaller number of characters employed made necessary the greater use of combination. The Greeks had a symbol for sixty or for ninety, but the Latins must place together several numeral signs of smaller value so that the combination would equal the total required In Greek 60 might be expressed by ξ', one character; in Latin by two, LX. The more complex the number the greater became the relative cumbersomeness of the method. In Greek 1863 could be expressed by ,αωξγ'; in Latin it would be MDCCCLXIII. The result of these cumbersome combinations was that with the Roman numerals it was virtually impossible to make calculations of any intricacy, and exceedingly difficult to make even simple ones. They might be employed for mere designation, as they are to this day used to express dates and to distinguish the pages of a preface; but for addition, subtraction, multiplication, division, and intricate arithmetical work, the Roman mathematicians were driven to use the Greek symbols and methods.

Meanwhile in the east other systems of numeral notation had been developed, some of which, in modified form, are in use there at the present time. The Babylonians were expert calculators; and the Chinese invented a notation which they still have. It was in India, however, that a system arose destined to supersede all others among civilized people.

There were probably some numerals in use among the Hindus a thousand or more years before our era, but no records exist earlier than the time of Asoka, in the third century B.C. From this time on occur inscriptions which contain some of the native number symbols. Two systems may be discerned, which possess respectively the characteristics of the Roman and the Greek.