In the introduction to the Loeb Greek Mathematics, Ivor Thomas writing in 1939 laments the fact that people no longer took joint classics and mathematics degrees. As he points out the achievement of Greeks in mathematics was remarkable and that the greatest period was after the death of Aristotle which for most classicist marks the end of the great period of Greek culture. Most Greek mathematics was in geometry. Algebra came later when major mathematical advances were being made in the Arab world.
If Theon of Smyrna had the simplest notion of algebra he could have supplied an easy proof for his observation below.
ἰδίως δὲ τοῖς τετραγώνοις συμβέβηκεν ἤτοι τρίτον ἔχειν ἢ μονάδος ἀφαιρεθείσης τρίτον ἔχειν πάντως, ἢ πάλιν τέταρτον ἔχειν ἢ μονάδος ἀφαιρεθείσης τέταρτον ἔχειν πάντως· καὶ τὸν μὲν μονάδος ἀφαιρεθείσης τρίτον ἔχοντα ἔχειν καὶ τέταρτον πάντως, ὡς ὁ δ , τὸν δὲ μονάδος ἀφαιρεθείσης τέταρτον ἔχοντα ἔχειν τρίτον πάντως, ὡς ὁ θ , ἢ τὸν αὐτὸν πάλιν καὶ τρίτον ἔχειν καὶ τέταρτον, ὡς ὁ λϛʹ [ἢ μηδέτερον τούτων ἔχοντα τοῦτον μονάδος ἀφαιρεθείσης τρίτον ἔχειν πάν–τως], ἢ μήτε τρίτον μήτε τέταρτον ἔχοντα μονάδος ἀφαιρεθείσης καὶ τρίτον ἔχειν καὶ τέταρτον, ὡς ὁ κεʹ.
There is something unique to square numbers that either they are divisible by three or they are divisible by three if you subtract one. Or again they are divisible by four or they are divisible by four if you subtract one. So a number divisible by four is divisible by three after subtracting one -for example 4. Again a number divisible by three is divisible by four if you subtract one – for example 9. Again the same number can be divisible by both three and four – for example 36. Or if it is divisible by neither three or four it becomes divisible by both three and four if you take away one – for example 25.
Theon of Smyrna (in Loeb Greek Mathematics Vol 1)
The basis of an algebraic proof of the above would be from the difference of two squares where
x^2 – 1 = (x + 1)(x – 1)
If x is not divisible by three then either x + 1 or x – 1 must be divisible by three so x^2 – 1 must be divisible by three. If it is divisible by three then x^2 must be divisible by three.
If x is not divisible by two then both x + 1 and x – 1 must be be divisible by two so x^2 – 1 must.be divisible by 4. If x is divisible by 2 then x^2 must be divisible by four.
You could go on to get either properties of squares and squares minus one by these means but we just want to prove Theon’s observation here.