This braid is found in the DMC Irish Crochet Lace book, by Therese Dillmont. It is Plate II, using figures 20, 50 and 75 to make it. If you go to the Antique Pattern Library look down under "D" for the pdfs. This braid is in pdf no. 2. The figures are in pdf no. 1.
The example on the left l
ooks like a tatting piece is crochet, and the one on the right looks to be machine made lace. All three designs share the repeated ring-in-a-circle motif, laid out in a linear fashion.