Resident in France, this highly unusual small lathe must have been designed by the same team responsible for Citroen cars of the classic era i.e. every single part breaks with convention to produce a remarkable machine arranged in such as way as to reinforce that well-known saying, "The Shock of the New".
Some guesswork is necessary to get to grips with this odd contraption but, probably made before the 1902s, the lathe might be categorised as a "semi-plain-turning" type i.e. without screwcutting but with the carriage driven along the bed by a hand-turned leadscrew held between the V and flat bed ways. Unfortunately, what we presume to be a leadscrew covered, so there is a slim chance that instead it's a drive shaft with connection to the saddle by a keyway driving through a worm and wheel assembly. Protruding from the face of the bed at the tailstock end was a crank handle, this being connected by bevel gears to a lower and shorter slave shaft parallel to the upper. The two shafts protruded through the bed's end face where they were joined together by a pair of gears - no doubt arranged so as to give a reduced ratio and so  provide a finer, more controlled feed. A similar arrangement of gears connecting a shaft and leadscrew was used on the Drummond flat-bed lathes of the early 20th century - though this much simpler system  included both a power shaft and a leadscrew used for screwcutting
Normally one needs to make only a perfunctory mention of a lathe's tailstock, but in this case, has anyone seen a locking-to-the-bed mechanism like the one employed? A lever, set longitudinally along the tailstock's centre line was drilled near its front end to carry an eye bolt attached to which was a clamp plate beneath the bed. At the rear of the lever was a vertical bolt that, when screwed down, pivoted the lever against its front end to lift the eye bolt and lock the tailstock down. The picture below should make the mechanism clear.
The compound slide rest assembly also failed to escape the clutches of the Rube Goldberg (USA) Heath Robinson (UK) designer. While the use of reduction gears on the cross-feed screw was not unique (though their exact form is unknown) the method of driving the non-swivelling top slide certainly was. It seems that the mechanism started with a cylindrical housing holding, at the top, a horizontal shaft turned by a crank handle. At the end of the shaft, a pair of bevel gears turned the drive through 90° to rotate a vertical shaft at the bottom of which must have been a second pair of gears to turn the drive through 90° - and so turn the horizontal feed screw.
The headstock was, of course, out of the ordinary. Lacking a slow-speed backgear, the spindle ran in an extraordinarily long front bearing secured by a 4-bolt cap with a much smaller bearing to the left. The original pulley would have been just a 2-step flat, now modified to take side-by-side V-belts. A long lever (which appears to be an original fitting) passed beneath the pulley - though its function is unknown.
Another puzzle is the purpose of a crank handle, presumably fitted to a screw, that emerges from the right hand side of the cross-slide's front face
The two small brass plates fixed to the front face of the bed cannot be read - but as they are not part of the casting, may well be by the supplying dealer, one being marked "Gambin". Hence, the country of origin is unknown, but probably France, as there appears, as in the UK, to have been a plethora of small makers (circa 1920 to 1950) turning out such machines. During the last ten years, an increasing number of lathes like this have been found in France - though unfortunately there is a lack of advertising literature featuring them. Should you be able to help with sales literature for lathes like this, the writer would be interested to hear from you.
Si vous êtes en mesure d'aider avec de la documentation commerciale pour des tours comme celui-ci, l'auteur serait intéressé à avoir de vos nouvelles..