When we turn to the social divisions we find in Domesday and other documents classes of society in these districts bearing purely Norse names, dreng, karl, karlman, bonde, thrall, lysing, hold; in the system of taxation we have an assessment by carucates and not by hides and virgates, and the **duodecimal** rather than the decimal system of reckoning.

The development of astronomy implies considerable progress in mathematics; it is not surprising, therefore, that the Babylonians should have invented an extremely simple method of ciphering or have discovered the convenience of the **duodecimal** system.

The value of 7r for **duodecimal** arithmetic is 3+1/12+8/122+ 4/12 3 +8/12 4 +..

Hence we see that it probably passed from the East through Greece to Etruria, and thence became the standard foot of Rome; there, though divided by the Italian **duodecimal** system into 12 unciae, it always maintained its original 16 digits, which are found marked on some of the foot-measures.

Calculations involving feet and inches are sometimes performed by means of **duodecimal** arithmetic; i.e., in effect, the tables of square measure and of cubic measure are amplified by the insertion of intermediate units.