  Following the programme set out in Part I of this work, we develop a
conceptual higher order differential calculus. The '' local linear algebra ''
defined in Part I is generalized by '' higher order local linear algebra ''.
The underlying combinatorial object of such higher algebra is the natural
n-dimensional hyper-cube, and so we qualify this calculus as '' cubic ''. More
precisely, we define two versions of conceptual cubic calculus: '' full '' and
'' symmetric cubic ''. The theory thus initiated sheds new light on several
foundational issues.
