In this first English book devoted entirely to tatting, the join as we know it has not yet been invented, so Mlle Riego overcomes this difficulty by using a netting needle or sewing needle instead of a shuttle. She made joins by passing the needle through picots. This method was soon supplanted by the regular join, but was a revolutionary step by allowing larger pieces to be made without laboriously tying the picots together with bits of thread.
In this book, the needle is manipulated the same way as a shuttle, so this is not needle tatting in the modern sense.
Mlle. Riego's description of the single stitch may seem baffling when you first read it, but take a shuttle in hand and follow along and it becomes clear. Once made, you can recognize this as our "second half" of the double stitch.
On page 6 she writes:
Compare to Mrs. Gaugain's instructions here. Do you think Riego's is clearer? It is a nice touch that she points out that the core thread should be able to slide. And, like all the others before, she includes the ubiquitous 20 half stitch edging.