Lathe: Cutting a Thread with Gears
The machine from Chapter 4.1 and How to Run a Lathe, running live. Pick a thread. The change gears swap, the lead screw turns at the true ratio, and the tool cuts the true pitch.
The one equation that cuts every thread
The carriage must advance $1/N_c$ inch per spindle revolution. The lead screw advances it $1/N_s$ inch per lead-screw revolution. The gears must supply the ratio:
$$ \frac{\omega_{\text{screw}}}{\omega_{\text{spindle}}} = \frac{N_s}{N_c} = \frac{\text{driver teeth}}{\text{driven teeth}} $$
Watch the accent spokes on the two gears. At 8 TPI they turn together. At 24 TPI the driven gear turns at one third speed, and the thread ticks land three times closer. This is the change-gear chart on the old headstock, derived. Chapter 4.1 gives the proof; How to Run a Lathe (1934) gives the practice.