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 no direct bearing on practical life could awake no interest. As a consequence, not only the higher geometry of Archimedes and Apollonius, but even the Elements of Euclid, were entirely neglected. What little mathematics the Romans possessed did not come from the Greeks, but from more ancient sources. Exactly where and how it originated is a matter of doubt. It seems most probable that the "Roman notation," as well as the practical geometry of the Romans, came from the old Etruscans, who, at the earliest period to which our knowledge of them extends, inhabited the district between the Arno and Tiber.

Livy tells us that the Etruscans were in the habit of representing the number of years elapsed, by driving yearly a nail into the sanctuary of Minerva, and that the Romans continued this practice. A less primitive mode of designating numbers, presumably of Etruscan origin, was a notation resembling the present "Roman notation." This system is noteworthy from the fact that a principle is involved in it which is not met with in any other; namely, the principle of subtraction. If a letter be placed before another of greater value, its value is not to be added to, but subtracted from, that of the greater. In the designation of large numbers a horizontal bar placed over a letter was made to increase its value one thousand fold. In fractions the Romans used the duodecimal system.

Of arithmetical calculations, the Romans employed three different kinds: Reckoning on the fingers, upon the abacus, and by tables prepared for the purpose.[3] Finger-symbolism was known as early as the time of King Numa, for he had erected, says Pliny, a statue of the double-faced Janus, of which the fingers indicated 365 (355?), the number of days in a year. Many other passages from Roman authors point out the use of the fingers as aids to calculation. In fact, a finger-symbolism of practically the same form was in use not only in