Addressing this topic, Reviel Netz said it was possible to use the comparative method to establish the archetype of these diagrams, although one could not be sure this archetype was the same as the original used by the ancient author.
The ancient Greek mathematical diagram was different from modern mathematical diagrams because it contained new information that was not in the text and it was schematic rather than pictorial (you could have straight lines represented by curves for instance–the thing was to get the configuration right, and it was topological rather than metric). It differed from ancient Babylonian and Chinese texts because there the text contained instructions relating to something external that was to be gridded, cut, folded and so on while the Greek diagram was part of the text.
The Greek mathematical text differed from the contemporary literary text because that was devoid of illustrations or other forms of articulation (word divisions, capitals, etc) and merely comprised one letter after another. This was because it was merely a blueprint with which one could reconstruct a speech act.
So how had the very schematic diagram originated? Perhaps it was by influence of the schematic uniformity of the written text, rather than some epistemological insight of the first Greek mathematicians. Thus the diagram became part of the text, and mathematics (geometry) shared in the prestige of literary culture.
To which I would comment that a combination of images and argument is surely characteristic of mathematical thought, and the fact one didn’t bother to sort it out may merely have meant a greater closeness assumed between author and reader. The diagram below (by Albert Einstein) after all doesn’t look very much like gravitational lensing…