Since the popularisation of techniques of decimal fractions at the end of the sixteenth century, western mathematics has drawn inspiration from the fluent manipulations of more and more general, more and more abstract, kinds of numbers.¹ I shall refer to mathematics that uses, like this, some system of numbers sufficiently general to include fractional quantities and their arithmetic as "arithmetised mathematics'. For example, Mesopotamian mathematics and astronomy is arithmetised; but we have no unambiguous evidence of the influence of this Mesopotamian mathematics on Greek mathematics before the second century BC; and thereafter, in Greek texts, Mesopotamian sexagesimal numbers are found only in astronomical contexts. I am exploring a novel interpretation of early Greek mathematics in which Mesopotamian influence on Greek mathematics is minimal, even non-existent, and will not consider this sexagesimal evidence further here.