How did Gauss conjecture there were nine Heegner numbers? Coming from someone not very knowledgable in algebraic number theory it seems odd. At the time they didn't have the computing power to determine whether very high values (>>163) were Heegner numbers; so, why even assume there was a finite amount rather than infinite (let alone exactly the nine there are)?

Gauss had plenty of computing power. He calculated class numbers up to 2000, and found they got scarcer as he climbed higher, with none at all after 163. That seemed enough to conjecture there weren't any more. 
Gauss was working with quadratic forms, rather than quadratic fields, and the bigger the discriminant, the easier it was to find inequivalent forms, so it stood to reason that eventually there would be no discriminants with just one class of forms. 
