Sum of cubes in Archimedes way

siu-kái

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.