A tweet by @Therese_LW on Sir Joseph Whitworth the Victorian engineer reminded The Lad [@isambardslad] of an interesting question. It sparked off an Twitter thread with @BillHarvey2
Whitworth was an engineer close to Brunel in any global ranking. He made enormous strides in improving the methods and accuracy of machine tools. In general, he greatly improved the efficiency of engineering manufacturing techniques. In this work he drove the Industrial Revolution into a higher gear.
Screw threads are an enormously important factor in engineering manufacture quite apart from their use in a multitude of types of fastener. In the form of lead screws they not only allow heavy masses to be moved but also, in many types of machine tools, allows the distance of movement to be very accurately controlled.
A lead screw on a simple lathe is an important example of this. Here a cutting tool is usually mounted on a saddle whose position is governed by the lead screw. Such a machine can, using the lead-screw as a controller and via gearbox in the headstock, cut another screw thread.

The point here though is that a new screw cut on such a lathe cannot have a better accuracy than the lead-screw that controls the cut. It can only be, at best, as accurate as the lead-screw or, to some degree, worse. This seems to be a completely general feature of machining.
So, this is the question. How can you get from a screw thread that looks, in a wine-press of the early 18thC, like this….

….to one that looks, and is, much, much more accurate like, for example, this lead-screw?

The Lad and @BillHarvey2 had an interesting, if rather compressed discussion on Twitter on the topic. He made the point that, in the early days, the toolmakers would have improved the shape and accuracy of a thread like the wine press or somewhat smaller, manually with hand tools like files and scrapers. This could then be a master screw to make others similar.
The Lad believed that there would be a limit to what could be done this way as the threads got smaller in diameter and the required accuracy increased. As was said above, you cannot expect to make an identical screw in a lathe that is better than the lead-screw you are using. It can only be as accurate or worse.
Then Bill Harvey gave what seems to be a breakthrough idea. A lathe is not limited to making screw threads that are the same dimensions as the lead-screw. So, let us fit our best, hand-improved screw as the lead-screw in a lathe. Then we set the lathe gearing suitably to cut a smaller diameter and finer [more threads per inch] screw. This way, the inaccuracies in the existing lead-screw will still be there but reduced in absolute terms in the new lead-screw. Fitting this new, finer and smaller screw back as a lead-screw in the lathe [probably a specially designed lathe for the job], we can then use it, with another gear change, to make a large screw again. This new one could be improved even more by hand.
Then this process could be repeated, with an improvement in accuracy each time, over several cycles of large to small to large threads. This is a boot-strap type process that seems to make possible an improved thread. There are other aspects perhaps such as a wide-span follower on the lead-screw as part of the improvement process.
All this so far though is deduction and an element of supposition based upon engineering judgement. The Lad must therefore check it if possible. First port of call will be “Sir Joseph Whitworth – The Worlds’s Greatest Mechanician” by Atkinson. See if that supports our deductions.
Thanks to @BillHarvey2 for the original discussion and the large/small thread idea and to @Therese_LW for raising the matter of Whitworth that sparked us off. Bill Harvey also posted on this subject.