  First we argue in an informal, qualitative way that it is natural to enlarge
space-time to five dimensions to be able to solve the problem of elementary
particle masses. Several criteria are developed for the success of this
program. Extending the Poincare group to the group C of all angle-preserving
transformations of space-time is one such scheme which satisfies these
criteria. Then we show that the field equation for spin 1/2 fermions coupled to
a self-force gauge field predicts mass spectra of the desired type: for a
certain range of a key parameter (Casimir invariant) a three-point mass
spectrum which fits the ``down'' quarks d,s, and b to within their experimental
bounds is obtained. Reasonable values of the coupling constant (of QCD
magnitude) and the range of the spatial wave function (a few fermis) also
result. Compatibility with the electroweak theory is also discussed.
