As usual, I'm two steps behind
Thunderbird, but it sounds like you have (actually your begonias have) Powdery Mildew:
Read here:
http://plantclinic.cornell.edu/FactSheets/powd ery/powdery.htm
It's probably too late in the season to due anything very helpful, however, be sure to thoroughly remove all leaf and plant residue when doing the fall clean-up.. Additionally, next spring, at the first sign of thaw (if your area freezes) spray the ground area with a product known here in the U.S. as
Dormant Oil. It's a foul smelling sulphur compound, but environmentally friendly. Don't get any on your clothes or shoes! Then, next season, if the mildew does re-appear, look for a product called GreenCure�, which is a potassium bicarbonate-based fungicide and is the safer alternative to harsh chemical fungicides. Also environmentallly friendly. A similar product is made from chrysanthemum seeds and contains Neem Oil... one brand we use is Green Light and is very effective... Best of luck!