The texts we have read so far come not from works extant under the names of Archimedes, Dionysodorus, or Diocles. They were handed down in a single work, extant under the name of a relatively obscure scholar: Eutocius of Ascalon. In the sixth century ad, Eutocius wrote a series of mathematical commentaries, of which one, the commentary to Archimedes' Second Book on the Sphere and Cylinder, is especially rich in mathematical and historical detail. Having reached Proposition four, Eutocius noted the lacuna in Archimedes' reasoning. He has (so he tells us) uncovered Archimedes' original text, which he then incorporated into his commentary. Finally, he added into it the solutions by Dionysodorus and Diocles. This, then, is our main source for the ancient form of the problem (we also happen to have the same solution by Diocles, preserved in Arabic translation).

Was Eutocius' work a mere record of the past, or did it make some original contribution to the history of mathematics? In this chapter, I argue that, already in the work of Eutocius, we can find mathematics making the transition from problems to equations. This comes at seemingly trivial moments, of little consequence in terms of their original mathematical contribution. Eutocius, without noticing this, occasionally happens to speak of mathematical objects that are rather like our quantitative, abstract magnitudes, and not the spatial geometrical objects studied by Classical mathematicians. He stumbles across functions and equations, without ever thinking about it.