Though its title /"Integral Geometry/" may appear somewhat unusual in thiscontext it is nevertheless quite appropriate, for Integral Geometry is anoutgrowth of what in the olden days was referred to as /"geometric probabil-ities./"
Originating, as legend has it, with the Buffon needle problem （which afternearly two centuries has lost little of its elegance and appeal）, geometricprobabilities have run into difficulties culminating in the paradoxes ofBertrand which threatened the fledgling field with banishment from the homeof Mathematics. In rescuing it from this fate, Poincar6 made the suggestionthat the arbitrariness of definition underlying the paradoxes could be removedby tying closer the definition of probability with a geometric group of which itwould have to be an invariant.