First part, to decompose each cubic into a sum of squares, then compose them from different direction.
Let
so
apply it to
...
To sum these n identities, let
where
Second part, make use of
; apply it to
:
...
To sum these n identities,
OK, easy part now, to merge the results of first part and second part, we can get
BTW, Archimedes would use
at first part, and use
at second part.