This article described one way to make the crescent-shaped piece by gluing together several pieces of veneer, in this case ash. The crescent-shaped piece made this way has proven itself in practice with high-power tests, and is also very strong and light. That said, I don't think the Greeks and the Romans would have used this method, because it takes much more effort than some of the other methods described in this Wiki
Unfortunately there are no photographs of the manufacturing process itself.
Making the bending form Edit
First make a bending form. This particular form is composed of two planks that form the sides, and a number of round dowels going through holes in the planks. The dowels follow the curvature of the to-be crescent-shaped piece. You should make the center part of the form perfectly flat for a significant length, so that the piece can attached directly to the end of the case without a separate adapter. The form is rigidified by a perpendicular beam at the bottom attaching the planks together. This beam also allows clamping the center of the crescent-shaped piece tightly. All the pieces are glued together.
Here are some pictures of the bending form, first from below:
From the side:
From the top:
And an overview from an angle:
The form could have equally well been made from a single, large block of wood.
Gluing the veneers into shape Edit
Once the bending form is ready, cut the veneer strips roughly to length. Note that you need to have about 5cm extra length at both ends, or you will have issues squeezing the ends of the strips against the form (been there). It is also a good idea to use veneers that are wider than necessary, so that if they slip or rotate a bit during gluing, you can simply plane the glued piece to correct form, instead of having to start from scratch.
Take two small pieces of wood and place them to the ends of the crescent-shaped piece, so that they are perpendicular to it: this will help the clamps apply even pressure along the entire width of the crescent-shaped piece. Glue these two pieces to the ends of the topmost veneer. Once the glue has tried, apply glue to all the veneer surfaces that are to be glued, then lay the veneers on top of each other and place them on top of the form.
Apply pressure with clamps to the blocks of wood at the ends of the topmost layer, ensuring that the veneers do not move too much sideways or length-vise. If necessary, clamp the piece from the middle also. You may need to use small blocks of wood under the form to prevent damage to the dowels.
Once the glue has dried, plane the crescent-shaped piece to correct form. Finish off with sandpaper. The end result should look like this:
As can be seen, the gluing job has been less than optimal:
It seems that only the outermost glue joint is a bad one. This is probably because there was uneven pressure on the topmost joint. The lower glue joints seem fine. If aesthetics are important, then you can simply rasp away the topmost layer and finish the job with sandpaper.
Making an adapter for the case Edit
You might not need an adapter if the middle part of the form is made entirely flat. However, in that case you need to clamp the middle of the piece from the center, and only after that clamp the ends. However, if the form is curved along its entire length, then you need to make an adapter for the case. The adapter is simply a piece of wood, which has a straight side and a curved side, with the curve matching the curvature of the crescent-shaped piece:
Attaching the crescent-shaped piece to the case Edit
As seen from the pictures above, I took the lazy way out and simply screwed the crescent-shaped piece into the case. The cheiroballistra manuscript seems to describe a rectangular mortise and tenon joint instead. That should work fine and look less out of place, if done properly.