The model name is 1R, nor R-1, according to the collet box labels. The collets are marked Rivett 1R. The collet threads are slightly different from common Webster-Whitcomb (WW) or B8 collets.
I have two of them, and both have the original cardboard box of collets with the lathe serial numbers (89 and 451) written in the box lid. Both have the round base. They have cast beds with a chrome-like color on the finished surfaces. I have a few extra 1R collets. The center height and bed shape allows slide rests and such from WW lathes to fit the Rivett. Number 451 has the three red knobs, but 89 has only the T-bolt red knob. Note the instructions say the lathe has ball bearings, not tapered roller bearings.
Lathe 451 was bought new by a talented professional engraver who apparently used watch lathes for a hobby. He bought the Rivett new and apparently barely used it. The lady who sold it to me around 1984 also had a gorgeous Derbyshire outfit that the engraver must have soon bought as a replacement for the Rivett. She would not sell the Derbyshire. I did get the engraving tools.
They are nice lathes, as would be true of any Rivett product. Apparently, the end of WWII produced a rush to watch repair schools by ex-soldiers with free tuition grants. There were a lot of new watch lathe designs that came out in that era and lots of lathes that show little use. Rivett may have priced theirs too high to compete, and the odd collet did not help. The other lathe makers kept with the WW collets or the less common but still popular Moseley design.
Rivett made watch lathes in the 1880's and later and those were high priced, with steel beds where all the competition used cast iron. The old Rivett watch lathe collets were very different from the WW standard. I have a very nice old Rivett watch lathe, but have only a few of the old collets for it.
I would sell the Rivett lathes and extra 1R collets.
Larry
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