After years of trying to turn rubber into something useful that wouldn't freeze rock hard or melt in the hot sun, Charles Goodyear was struggling. He'd been experimenting for years and invested everything he owned in rubber research, but hadn't been able to create a commercially viable product, and his family was starving.
But things started to turn around. First, he poured some nitric acid onto some rubber that had been colored gold to remove the color. It turned black, so he threw it out, but removed it from the trash when he realized that it had become hard on the outside, and was smoother and drier than any previous rubber. But it still melted in high heat.
He started using sulphur in his experiments, and here's where things get a little murky. As the story goes, in a fit of excitement, he tossed some rubber that had been treated with sulphur up in the air, and it landed on a stove. But instead of melting, it charred, creating an almost leathery, heat-resistant waterproof substance.
After further experimentation, he realized he could get the most effective results by using steam to heat up the mixture of rubber and sulphur he'd created. Finally, he found success.
Goodyear vehemently disagreed with those who label this finding an accident, since he's the one who followed through with it all. But (if the story is true), the discovery still depended on one lucky accident.