Does mathematics have a history? I believe it does, and in this book I offer an example. I follow a mathematical problem from its first statement, in Archimedes' Second Book on the Sphere and Cylinder, through many of the solutions that were offered to it in early Mediterranean mathematics. The route I have chosen starts with Archimedes himself and ends (largely speaking) with Omar Khayyam. I discuss the solutions offered by Hellenistic mathematicians working immediately after Archimedes, as well as the comments made by a late Ancient commentator; finally, I consider the solutions offered by Arab mathematicians prior to Khayyam and by Khayyam himself, with a brief glance forward to an Arabic response to Khayyam.

The entire route, I shall argue, constitutes history: the problem was not merely studied and re-studied, but transformed. From a geometrical problem, it became an equation.

For, in truth, not everyone agrees that mathematics has a history, while those who defend the historicity of mathematics have still to make the argument. I write the book to fill this gap: let us consider, then, the historiographical background.

My starting point is a celebrated debate in the historiography of mathematics. The following question was posed: are the historically determined features of a given piece of mathematics significant to it as mathematics? This debate was sparked by Unguru's article from 1975, “On the Need to Re-write the History of Greek Mathematics”.