Special relativity was formalised around ten years earlier than general relativity (around 1905), which better fits in with my mental timeline of the fin de siecle.

I can't see what special relativity would have to do with Hume. It just extended the principle of relativity, which was already introduced by Galileo, to the propagation of light at a finite speed, though with all kinds of counter-intuitive results such as the relativity of simultaneity. By itself, it still doesn't predict (say) gravitation. (It does predict conservation of energy, momentum and angular momentum if you assume space-time is homogeneous and isotropic and use Noether's theorem, but so does Galilean relativity for that matter.)

On the other hand, general relativity, from a small number of very simple assumptions, predicts quite a lot of things (pretty much any non-quantum phenomenon which had observed back then except electromagnetism). Indeed Einstein said he was completely certain his theory would prove to be true before it was even tested. EDIT: you actually need more data than I remembered to get to GR: see http://lesswrong.com/lw/jo/einsteins_arrogance/757x

(Wow, now that I'm trying to explain that, I realize that the difference between SR and GR in these respects are nowhere near as important as I was thinking.)

Anyway, there's still no logical reason why those very simple assumptions have to be true; you still need experience to tell you they are.

The comments to http://lesswrong.com/lw/jo/einsteins_arrogance/ go into more detail about this.

If the former, I think it was more a widespread idea that the majority of the educated shared, rather than one person's assertion.

Can you give me some pointers? I can't recall ever hearing about that before.