  We construct a consistent reduction of type IIA supergravity on S^3, leading
to a maximal gauged supergravity in seven dimensions with the full set of
massless SO(4) Yang-Mills fields. We do this by starting with the known S^4
reduction of eleven-dimensional supergravity, and showing that it is possible
to take a singular limit of the resulting standard SO(5)-gauged maximal
supergravity in seven dimensions, whose eleven-dimensional interpretation
involves taking a limit where the internal 4-sphere degenerates to RxS^3. This
allows us to reinterpret the limiting SO(4)-gauged theory in seven dimensions
as the S^3 reduction of type IIA supergravity. We also obtain the consistent
S^4 reduction of type IIA supergravity, which gives an SO(5)-gauged maximal
supergravity in D=6.
