[alancalverd (OP)]

So I'm cutting a screw thread on a lathe. The lead screw turns and moves the tool along the workpiece, and the gear ratio between the chuck and the lead screw determines the pitch of my product screw. An everyday occurrence in thousands of workshops all over the world.

But the lead screw needs to be at least as precise as the one I'm making.

So how did they make the first lead screw?

[Bored chemist]

But the lead screw needs to be at least as precise as the one I'm making.

Not if you are clever.

Anyway there are a number of ways of making a pretty good leadscrew; an easy one is to wrap a wire round a rod.

[evan_au]

They used nails and rivets for a long time...

Apparently, screws made of wood were used for pressing wine in ancient Greece.
Archimedes introduced this design...
https://en.wikipedia.org/wiki/Archimedes%27_screw

But screws made of metal for joining wood went into mass production in the 1780s.
https://en.wikipedia.org/wiki/Screw#History

[alancalverd (OP)]

So far, then, we have one suggestion that requires the existence of precisely formed wire and a likewise precise circular rod, the manufacture of neither being dependent on the existence of a lathe, and the suggestion that woodscrews, which look nothing like lead screws, were made by machines that employed leadscrews which had been formed by magic in the 18th century.

Come on, guys, this is a simple chicken and egg question: how did whoever made it, make the first leadscrew with sufficient precision to make all the others?  It's a question I've been meaning to ask  since "O" level metalwork!

[vhfpmr]

A winch and cable will give you linear motion from rotary.

[Bored chemist]

A winch and cable will give you linear motion from rotary.

That's a different clever answer from the one I was told about (see below)
I'm now waiting to Alan to misrepresent that as saying that this sort of thing was available in the stone age or some such tosh.
https://www.machinemart.co.uk/c/hand-electric-winches/

leadscrews which had been formed by magic in the 18th century.

Not by magic; by clever people.

Clever enough to realise that if you make the nut that engages with the lead screw out of a tube lined with leather, the soft leather will average out the imperfections in the leadscrew and let you cut a better one.
After a few iterations you have a really good one.

You don't need a leadscrew to cut a cylinder on a lathe nor do you need a lathe to drill a hole in a plate to make a wire drawing die.
(Indeed, while I said that wire would give you quite a good thread, a bit of varnished string would do the job.)
So; as is always the case, no magic was required; just ingenuity.

[alancalverd (OP)]

A winch and cable will give you linear motion from rotary.

Linearish. I smell a physicist, using perfectly  circular cylinders and absolutely homogeneous inelastic  (but infinitely flexible) strings of infinitesimal thickness.

BC's leather averager rings a bell in the dim recesses of my memory. Now that's clever, and real-world engineering.

[Bored chemist]

Linearish.

A bit like cutting a cylinder on a lathe.
The further you are from the headstock, the more the workpiece bends under the load from the cutting tool.
Even if you deliberately try to cut a taper to compensate for that  you can't quite win because the correction isn't linear.
If you want really good precision, you need a feedback path. You have just invented CNC machining.

However, cutting a very good cylinder is a reasonable prospect.
If your cable is thin compared to the diameter of the winch, the diameter isn't very important.
If you put the winch a long way from the attachment point the fact that the cable is at an angle doesn't matter much.
You can, of course improve it significantly by... mounting the winch on a lead screw.
Or, you could use a second winch to move the main winch along axially.
There may be a reason why they use screws.

[alancalverd (OP)]

The further you are from the headstock, the more the workpiece bends under the load from the cutting tool.

Not if you use a tail center and the slideway is perpendicular to the headstock. If anything the taper will be outwards because the cutting tool will wear slightly, so you cut marginally oversize then skim backwards towards the headstock with a finishing tool.

[Bored chemist]

Even if the ways and the other bits and pieces were infinitely rigid and perfectly perpendicular, the workpiece would still bend because you push on it to get the cutting edge to go into the metal.
To a good approximation, the tool pressure is the same along the length of the cut, but the deflection isn't.
Using a tailstock helps.
A follow rest is (in principle) even better.

[alancalverd (OP)]

But if you push the tool so that the middle of your workpiece bends, the tool will keep cutting and the piece will spring back until the tool no longer touches the piece.

[Bored chemist]

But if you push the tool so that the middle of your workpiece bends, the tool will keep cutting and the piece will spring back until the tool no longer touches the piece.

Are you talking about a failure of Newton's third law, or a material with an infinite young's modulus (but still machinable)?

Realistically, machinists take what's called a "spring cut" to minimise the effect.
But that doesn't eliminate it.
If you were right they wouldn't need to so... why do they do it?

[evan_au]

You have just invented CNC machining.

North Korea discovered CNC machining - and wax lyrical about it (the chorus is recognizable by English-speakers, and starts at 24 seconds).
- A decade ago, this was one of their most popular songs...

[Petrochemicals]

Probably with a file and a file guide.

I doubt master screws are manufactured from other screws, probably a gear arrangement?

[Petrochemicals]

Centerless grinding does not rely on centres.

https://en.m.wikipedia.org/wiki/Centerless_grinding

[Bored chemist]

Centerless grinding does not rely on centres.

https://en.m.wikipedia.org/wiki/Centerless_grinding

Let's see you use that to create a thread.

[Zer0]

What about creating a metal Threading & Tapping tool n using it on Wood?

Then using the wooden Shaft & Nut to make a Metal lead screw & nut?