That’s exactly what I was wondering.
The change gear method is the traditional way - there is no possibility of getting “out of sync” because of the gear teeth. I think the modern cog belts / wheels have proven to be very reliable, just look at all the timing belts out there...
But first one needs to have a cutter...
The basic principle of the EUREKA attachment is to have a stationary tool and the work “lobes” in and out of the cut. Traditional relieving lathes are the opposite: the tool moves on the X-axis and the work rotates concentrically with the lathe’s Z-axis. The number of “reliefs” per rev is a function of the number of teeth the cutter has.
I am contemplating using the taper attachment mechanism (mostly the telescoping leadscrew) via a timed cam to move the cross slide. I would need the cam to rotate at the number of cutter teeth per one spindle rev. The cam profile would control the relief profile and the stroke length. This way, one could make cutters of various sizes & relief angles. Furthermore, if the cam drive was via a telescoping cardan shaft, one could use the lathe carriage and set up a desired Module or DP and make a gear hob. Gear hobs have straight flanks - eliminating the need to try and approximate the involute curve. And you only need one for all the gears, regardless of # of teeth.
View attachment 12953
Another thing, if the drive train to the cam was via a planetary, one could drive the ring gear “slower“ than the planet carrier and get a differential function. This would allow for helical cutters as long as the differential per rev is a function of the helix angle. (differential division is the mechanical principle behind an universal dividing head: with each rotation of the crank, the reference point moves backwards ((or forwards)) by the “differential amount” to allow the circle to be divided by a prime number, for example, without using a prime number hole circle plate [like 127]). I digress.
Maybe all this could be done on a Menziken M200? Hint, hint.... Joking aside, I think they fell victim to the CNC revolution, like so many other manufacturers of high quality manual machines.