>What was the old story?

I don't know, I was wondering what it was too. I've never seen anything that specifically reminded me of it (for example one of Aesop's fables) but I'd be surprised if it didn't originate in an oral tradition, it is hard to imagine that the idea of a spider luring prey into the web is new. As Bill says it could be a nursery rhyme but I can't remember any specific one. I'll keep looking.