We have the best evidence (e.g. in Theon of Smyrna) of the practice of representing square numbers and other figured numbers, e.g. oblong, triangular, hexagonal, by dots or signs arranged in the shape of the particular figure. (Cf. Aristotle, Metaph. 1092 b 12).
Thus, says Treutlein, it would be easily seen that any square number can be
turned into the next higher square by putting a single row of dots round two
adjacent sides, in the form of a gnomon (see figures on next page).
If a is the side of a particular square, the gnomon round it is shown by simple
inspection to contain 2a + 1 dots or units.
...
I think Treutlein's hypothesis is shown to be the correct one by the passage in
Aristotle's Physics already quoted, where the reference is undoubtedly to
the Pythagoreans, and odd numbers are clearly identified with gnomons
"placed around 1." But the ancient commentaries on the passage make the matter
clearer still. Philoponus says: "As a proof ... the Pythagoreans refer to what
happens with the addition of numbers; for when the odd numbers are successively
added to a square number they keep it square and equilateral.... Odd numbers are
accordingly called gnomons because, when added to what are already squares, they
preserve the square form.... Alexander has excellently said in explanation that
the phrase ' when gnomons are placed round' means making a figure with
the odd numbers ... for it is the practice with the Pythagoreans to
represent
things in figures.
pag367 Heath vol1
wp/Theon_of_Smyrna (ed. Hiller)
domanda che sorge spontanea in ElementsBook2 sull'algebra geometrica.