  We propose the possibility that compact extra dimensions can obtain large
size by higher dimensional inflation, relating the weakness of the actual
gravitational force to the size of the observable universe. Solution to the
horizon problem implies that the fundamental scale of gravity is smaller than
$10^{13}$ GeV which can be realised in a braneworld framework for any number of
extra dimensions. However, requirement of (approximate) flat power spectrum of
primordial density fluctuations consistent with present observations makes this
simple proposal possible only for one extra dimension at around the micron
scale. After the end of five-dimensional inflation, the radion modulus can be
stabilised at a vacuum with positive energy of the order of the present dark
energy scale. An attractive possibility is based on the contribution to the
Casimir energy of right-handed neutrinos with a mass at a similar scale.
