Guillaume Cornelis van Beverloo a.k.a. Corneille (2)

Researching Corneille for last Friday's post, I came upon these interesting pictures.

Corneille wearing a type of peeked cap with a txortena (the 'wick' in the center, which is the last bit of thread used to "close" the beret). 

I have no information on these hats, despite many hours spent searching. I do have a few similar hats in my own collection though, but apart from the label stating they are 'Basque' and 'made in France', they don't tell me anything. 

I would love to hear from readers who know more about these caps.