A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

THE TREATISE ON POLYGONAL NUMBERS 517
The last proposition, which breaks off in the middle, is
Given a number, to find in how many ways it can be
polygonal.
The proposition begins in a way which suggests that
Diophantus first proved geometrically that, if
8P(a-2) + (a-4) 2 = {2 + (2ti-1) (a- 2)
2
,
then 2P = n {2+ (n— l)(a— 2)}.
Wertheim (in his edition of Diophantus) has suggested a
restoration of the complete proof of this proposition, and
I have shown (in my edition) how the proof can be made
shorter. Wertheim adds an investigation of the main problem,
but no doubt opinions will continue to differ as to
whether Diophantus actually solved it.
XXI
COMMENTATORS AND BYZANTINES
We have come to the last stage of Greek mathematics ; it
only remains to include in a last chapter references to commentators
of more or less note who contributed nothingoriginal
but have preserved, among observations and explanations
obvious or trivial from a mathematical point of view,
valuable extracts from works which have perished, or
historical allusions which, in the absence of original documents,
are precious in proportion to their rarity. Nor must
it be forgotten that ! in several cases we probably owe to the
commentators the fact that the masterpieces of the great
mathematicians have survived, wholly or partly, hf the
original Greek or at all. This may have been the case even
with the works of Archimedes on which Eutocius wrote commentaries.
It was no doubt these commentaries which
aroused in the school of Isidorus of Miletus (the colleague
of Anthemius as architect of Saint Sophia at Constantinople)
a new interest in the works of Archimedes and caused them
to be sought out in the various libraries or wherever they had
lain hid. This revived interest apparently had the effect of
evoking new versions of the famous works commented upon
in a form more convenient for the student, with the Doric
dialect of the original eliminated; this translation of the
Doric into] the more familiar dialect was systematically
carried out in those books only which Eutocius commented
on, and it is these versions which alone survive. Again,
Eutocius's commentary on Apollonius's Conies is extant for
the first four Books, and it is probably owing to their having
been commented on by Eutocius, as well as to their being
more elementary than the rest, that these four Books alone