It's an "urban legend" of old tooling. A lathe cannot reproduce itself. At the very least its not going to make the ways, the headstock or tailstock casting, or the chip trays. But there are many, many other components which would be heavily problematic AT BEST to reproduce.
-It's not a 'legend', it's a concept.
The idea isn't that you take, say, a Southbend Heavy 10, and use it to produce
another Southbend Heavy 10. It's the concept of it- the lathe is the simplest and most basic machine tool. Don't think "modern variable speed lathe with quickchange gearbox", think "primitive wood lathe".
If you had nothing but some very basic woodworking tools (knife, axe, saw, etc.) you could make a primitive wood lathe simply with two points and a bow. Using that "lathe", you can make more precise round parts- such as an actual spindle. Using
that more precise lathe, you can make more and better parts- including, with work, actual threads for a leadscrew.
A primitive mill on the other hand, would first require the lathe.
Yes, the phrase is perhaps better said as "the lathe is the
simplest machine that can be used to produce another version of itself", but that doesn't quite roll off the tongue as easily.
And, if you want to argue about it, history shows the lathe came first- there were primitive bow lathes back in Egyptian times, whereas the first milling machine- developed, by the way, from a lathe headstock and shortened lathe bed- didn't make it's appearance 'til the early 1800s.
Doc.