Providence, Rhode Island, was home to many makers of machine tools with grinders by Leach and the Builders Iron Foundry and several lathe makers, including E. A. Eddy, E. A. Adams, Adams Bros, the Diamond Machine Co.; E.B. Bosworth, Charles A.Mann, J.M.Parker's Gage, I.H.Spencer and Narragansett. Unfortunately, apart from Diamond, little is known about the historical background of any others, the record being, for the moment, blank.
Very similar in appearance to the E. A. Adams, Providence, Rhode Island, and the E. A. Eddy, Providence Rhode Island, the Bosworth was of approximately 4-inch centre height and 10-inches between centres. It had neither backgear nor screwcutting and the wide, flat-topped V-edged bed (made from good-sized box-section casting) carried a single lever-action cross slide driven through a toggle mechanism. Interestingly, the E.A.Adams had a proper compound slide rest and its entirely possible that this was an option on the Bosworth, a hole being provided through the saddle casting to take a cross-feed screw.
Unlike the unusual E. A. Adams (which may, in any case, have been modified) the Bosworth was arranged conventionally with the main mass of the bed under the headstock and carriage and the tailstock end cantilevered out to the right.
With just two wide steps on its cone pulley, the headstock looks to have been constructed for industrial use, this arrangement being common on lighter and bench precision lathes of the time- a similar 2-step arrangement being used on the headstocks of the W.H.Nichols and (three-bearing) Ames type. Threaded on their ends for adjustment and drawn down into their tapered seats by large ring nuts, both headstock bearings were mounted in a light casting with that on the left held on the end of a slender arm cantilevered out to the left and that on the right on top of a short, un-braced vertical post. 
It appears inconvertible that, as all three lathes appear to be identical in both layout and detailing and come from the same area, all must be by the same makers and badged, as required, by the customer for sale into the trade.
Mark Mason, in the USA, has discovered that Bosworth has been a well-known name in Warren, RI business circles for a considerable  time and that both E. A. Eddy and E. A. Adams were in the machinery and jewellery business, both as manufacturers and as resellers, with E. A. Adams & Sons surviving until as late as 1982. A web search revealed a number of obituaries and resumes of ex-employees, so the likelihood is that this was a considerable business; in the Providence Jewelers Circular, of 4 Aug 1920, it  says: The EA Eddy Machinery Co has purchased the manufacturing jewellery plant of the HA Kirby Co 85 Sprague St and will sell the tools machinery etc in lots to suit purchasers: and The offices of the EA Eddy Machinery Co have been removed upstairs in their building corner of Clifford and Eddy Sts thus giving additional space for the display of jewellers and general machinery on the lower floor.
Should any reader own an E. A. Adams lathe, (or an E. A. Eddy or Bosworth)  the writer would be interested to hear from you..