  Integration-by-parts (IBP) identities and differential equations are the
primary modern tools for the evaluation of high-order Feynman integrals. They
are commonly derived and implemented in the momentum-space representation. We
provide a different viewpoint on these important tools by working in
Feynman-parameter space, and using its projective geometry. Our work is based
upon little-known results pre-dating the modern era of loop calculations: we
adapt and generalise these results, deriving a very general expression for sets
of IBP identities in parameter space, associated with a generic Feynman
diagram, and valid to any loop order, relying on the characterisation of
Feynman-parameter integrands as projective forms. We validate our method by
deriving and solving systems of differential equations for several simple
diagrams at one and two loops, providing a unified perspective on a number of
existing results.
