  We formulate D=11 supergravity over the octonions by rewriting 32-component
Majorana spinors as 4-component octonionic spinors. Dimensional reduction to
D=4 and D=3 suggests an interpretation of the so-called 'dilaton vectors',
which parameterise the couplings of the dilatons to other fields in the theory,
as unit 'octavian integers' - the octonionic analogues of integers. The
parameterisation involves a novel use of the duality between points and lines
on the Fano plane, and suggests a series of consistent truncations with
N=8,4,2,1, giving the 'four curious supergravities' studied by Duff and Ferrara
