I had always assumed that the horse chestnut had something to do with the horse radish, and I'm therefore terribly sad to find that I was, essentially, wrong. I had had a lovely picture in my mind of a proud horse tending his garden.
Horse chestnuts were once used as a medicine for horses, they were believed to cure their coughs. Other horse medicines included horse bane (phellandrium equaticum) which cures horses with palsy, and horse cassia (cassia marginata) which helps with their constipation.
However, horse radishes don't cure a horse of anything at all, they're just big. A horse ant is a big ant. A horse cucumber is a big cucumber. A horse mushroom is a big mushroom. And a horse radish is a great big radish.
Incidentally, radish means root etymologically speaking, from the Latin radix. That's why, if you want to change things from the roots upwards, you are a political radical. This means that, so far as I'm concerned, all radishes are radicals and all radicals are radishes.
Or maybe I'm wrong.
P.S. If you want to hear me on the radio last night, you can follow this link. I start about half an hour in. I haven't listened to it myself, but that's because I have a pathological hatred of my own voice.