By the end of the first century BC, Rome had turned from a republic into an empire, and, in the course of the first two centuries AD, it secured control over most of Europe, North Africa, Egypt and the Near East. The empire is a constant presence in our evidence from this period, and it enters mathematical discourse in several ways. Managing an army, collecting taxes, keeping a census on such a vast scale implied centralized administrative practices (accounts, tax rolls, land surveys). Mathematics was also used to articulate views about politics, society and morals. It would be impossible to describe our period in a few words: let us just say that the world had become even larger than after Alexander’s expedition, exchanges of all types increased; and the textual past kept accumulating in the form of books and libraries. Turning to the evidence, apart from the usual survey of material sources, there are individual sections on Vitruvius and Hero. For the rest, authors have been assigned to the two sections ‘Other Greeks’ and ‘Other Romans’ on the basis of the language they worked in – geographically, they come from all over the place and they all belonged to the same Empire. A more earnest exploration of the Greek/Roman divide will be taken up in chapter 6.