Is it neighbourly to offer to pay HALF for a new fence even if the fence does not belong to us?

We got a letter in the post this weekend from our neighbour. Two of the panels have been falling over, and they have decided to replace the whole fence, and not just the two panels that were broken. The fence has been an eye sore ever since the day we moved in. We first thought it belonged to us, but then our other neighbour put us straight when our fence fell into his yard. (doh!, and insisted we fix it – well he helped me).

Anyway, I went out into the garden yesterday, and our neighbour was in the process of tearing down the fence. I offered to help him and in a goodwill gesture, offered to pay for half the new fence. The guy was totally shocked, although I thought it would be only fair to pay half, as it would benefit us both if we ever decided to sell our properties.

I think he thought "Mad South African"… but what do you think?

"Mad South African!!" or "the right thing to do?".

Who owns the fence to the right of your property if you are standing with your back to your house, looking down the length of your garden? 
My neighbour to the right tells me the fence to his left belongs to me. Here is a link on Yahoo that I found, and it seems that no one knows this answer?

I also found these on the Garden Law website: