Once enjoying  the position of producing more machine tools than any other American state, Ohio had, for the first half of the twentieth century, by far the greatest number of large, medium and smaller firms in the trade and produced enormous quantities of all types of metal working equipment. In 1905, for example, it manufactured well over three times as many lathes as its nearest competitor, Massachusetts, and also led in the manufacture of boring and drilling machines, planers, shapers, slotters, threading machines and many smaller specialised items; only in milling machines was it out performed by Rhode Island - a situation that was to be satisfactorily reversed in favour of the Ohioans a few years later.
Typical of the sort of enterprise which flourished in those early years, the Hamilton Machine Tool Company of Hamilton was founded in 1892 by its first president,  Mr. Charles F. Hilker, in premises near the junction of Market Street and Monument Avenue. Although the first product was a simple drill press, made in just one size, this proved so popular than, even in the financially insecure year of 1894, Mr. Hilker had generated sufficient confidence in his ability that he was able to raise funds sufficient to erect a new 60 x 200 feet building in Lindenwald - so greatly expanding his production capacity. By 1896 a much fuller range of drills was being sold and, in 1898, Mr. Hilker bought the Belmer Machine Tool Company of Cincinnati and combined their extensive plant with the Hamilton works. Soon the product range expanded to include a wider range of machine tools - radial-arm drills, shapers and planers - with company also becoming increasingly well known for their soundly built engine (centre) and simple capstan lathes of 14, 16, 18, 20, 22, 24, 26, 28, 32 and 36 inch swing. Thus, from humble beginnings, an organisation was created that, at the height of its prosperity in the early part of the 20th century, employed over 3000 people.
Hamilton screwcutting lathes were produced in two versions that the manufacturers designated as "Style A" and "Style B". The A had an early form of screwcutting gearbox, of a most unusual design, built under licence from Edward Flather whose machine tools were well respected in the latter part of the 19th and early part of the 20th centuries - the Style B used ordinary demountable changewheels.
Carrying a patent date of October 15th, 1901 lathes with the new gearbox were quite unlike any other in use at the time However, Hamilton did not recognise the Flather connection in their advertising, preferring instead to call the device the "Universal Screwcutting and Feeding Arrangement".  In so far as the drawings indicate, the mechanism consisted of a circular case containing a large ring gear in the form of a short "annulus cylinder" with teeth formed on its inside and outer walls. The inner gear could mesh, one at a time, with eight gears of different sizes arranged around its periphery. The shaft upon which each gear ran protruded through the outside face of the casing allowing the operator to grasp it and so engage and disengage the gears. The outer gear then drove a train of three gears (arranged below and to the side) connected to a reversing bevel box - from where the drive was taken to the leadscrew. Although a very compact design, able to cut 48 different threads from 1 to 56 per inch, contemporary criticism (which was not born out in practice) centred on the fact that, despite being well made from good-quality materials, the reliability and durability of such a mechanism might be questionable. Its complicated method of operation might also, it was claimed, have confused the operator and led to slower work and more mistakes. In the event, the design did not prove a success and the increasingly popular and reliable USA "Norton" quick-change gearbox continued to increase its market share. As a point of interest, the Norton box was not the first of its type, a similar arrangement of gears, of different sizes, placed in a "cone" on a common shaft, having been patented in 1868 by Humphreys. However, the Norton box, helped to class-leading prominence by the Hendey Company who first fitted it to their lathes in 1882, proved to be a long-term success story and it is still used today ijn irs original and modified forms.
If you a Hamilton lathe with the "circular" gearbox, the writer would be very interested to hear from you..

26" x 120" Type A with the (Flather patent) Hamilton "Universal Screw Cutting and Feeding Arrangement". The circular gear case, with its protruding selector levers, can be seen at the end of the lathe; the circular casing below it contained the train of gears which drove to the feeds reversing bevel box at the left-hand end of the leadscrew.

36" x 120" screwcutting lathe with changewheels and two types of tool slide - swivelling (shown mounted) and a presumably more rigid "fixed-position" example lying on the floor in line with the faceplate. Note the genuine "triple-geared" headstock with a conventional "backgear" mounted in front of the spindle with a second reduction taken from a point a little way along from its larger gear; a third reduction in ratio was created by the final drive to a gear cut on the inside rim of the faceplate. Some makers claimed "triple-backgear" status for lathes that had only two gear-driven reductions - their argument being that lathes with no backgear at all were often referred to as "single-geared", hence a conventional "single" installation of a backgear should be called double-geared and one with two backgears awarded the status of "triple".

16" x 72" Hamilton Type B lathe with screwcutting by changewheels and a separate power sliding and surfacing feed from a 4-step flat-belt drive pulley on the end of the powershaft.

End section showing the disposition of the
8 ring gears and the lower set of compound gears.

Side section through the screwcutting gear box and leadscrew-reversing bevel box (17).  No. 18 is the screwcutting leadscrew, No. 19 the feed shaft for the power sliding and surfacing and No. 20 the rod that provided an automatic knock off for the feeds.

Joe Michaels, writing from the United States in the year 2000, recounts an interesting story about using a Hamilton lathe:
I read the information on your web site about HAMILTON lathes and remembered - because I found it to be a radically different machine - the experience of running one about 20 years ago. The lathe was perhaps 12" swing (6" centre) x 48" between centres and appeared to be a toolroom-class machine.
The situation was as follows: I was working as an erecting engineer for a used engine, generator & turbine dealer. This dealer had sold an ex-US Navy combustion turbine to a small family-owned electric utility known as (if I remember rightly) "North-western Wisconsin Electric Company". They ran a variety of older generating units consisting of smaller low-head hydro turbines from the 1920's, some medium-speed diesels (a lot were salvaged from marine or locomotive applications) and had to acquire 5 Mw of power generation in a hurry as they were part of the Rural Electrification Administration's Program. They were in a very remote part of Wisconsin- with the nearest cities of any size being Duluth, Minnesota & Superior. The gas turbine plant was out in the woods, with a Native-American cemetery (complete with totem pole) on the access road.
The gang at Northwest Wisconsin Electric Company was used to buying anything and everything and making it work. As a routine thing, they bought old, large, synchronous motors (1500-2000 HP) from iron-ore processing mills and used them as generators; they'd add a thrust bearing and make new end bells if the motor had to go on a vertical hydro unit - having a shipyard shop up at Superior do the machining after they did the fabrication work in a barn. Their power plants were heated with woodstoves - all welded out of scrap steel plate and fired with timber cut along power-line right-of-ways. As I recall, they had one older plant operator, "Barney" who had one arm; they had rigged up a lot of paddles on various controls so this guy could take units on and off line (including manually synchronizing them). The story was that this guy had been a trapper, had been working his trap line on snowshoes and (the reason for the shooting varies here) got shot with a load of 00 buckshot in his arm. He tied a tourniquet on what was left of his arm and packed several miles on snowshoes to the nearest settlement (a bar room and general store - quite unchanged when I worked there). His forearm and elbow were amputated on the bar - the fellow showed me the spot where the surgery had occurred over a couple of shots and beers. Another principal in the company had claw-type prosthetic hands- having lost them at age 16 doing some hot line work to help his dad and uncles. This fellow went on to get an electrical-engineer's degree and worked as an erector for Allis-Chalmers before coming home. In short, they were an old-time gang of characters of a type which appears to no longer be in production ….
I had driven out to them in a 1980 International Harvester Scout; this is an extinct beast with a 4 ton winch on the front, mounted in a pressed-steel channel bumper assembly. I wanted to fit a winch guard to protect the fairleads and this guard had to have a pair of steel blocks machined with an angle matching that of each side of the bumper, so as to make a mounting surface parallel to the face of the winch. Weekends were slack time, so the folks there gave me the run of things; I found some scrap steel, veed it out and welded it up to a sufficient thickness to make the winch-mount block guards. I figured I would grind the blocks to dimension with an "angle grinder" as used by welders and boilermakers - as that was what most "good-old boys" would do in the situation. The fellows at the Utility Company drew me a map and gave me the keys to their shop in a little place called Grantsburg, Wisconsin. They told me their shop wasn't much, but that I could make free with it. I drove to Grantsburg and let myself into what had been a tin shed; inside was a Hamilton lathe with a four-jaw chuck and an old Garvin (or so it looked like) milling machine.
I set the blocks up in the Hamilton lathe and faced them to overall dimension - an excellent way to kill a rainy Saturday far from home. The Hamilton lathe was in beautiful condition; it had the circular quick-change mechanism you describe and I do not think it took me more than ten minutes to figure out how to use it. I found the mechanism easy to operate whilst, strangely, these new-fangled inch-metric quick-change gearboxes on European, Chinese or similar lathes give me a real problem in working out the feeds and threads. We have a 25" South Bend/Nordic lathe of 1972 vintage at the power plant where I work - and I can tell you that I spend longer trying to figure out my feeds or threads on this so-called "modern" machine.
As I recall, the little Hamilton lathe worked quite easily - with a little rolling of the spindle by pulling the belt you could line up the gears. I liked the Hamilton lathe and asked to buy it, but was told it was not for sale; for all I know, it may still be up there in that barn in Grantsburg, Wisconsin.
As far as Flather lathes go, I recall seeing one in a scrap yard back in 1973. That lathe had the regular Norton style quick-change box on it. It was part of an ancient machine shop which was being broken up. The lot included a Whitcomb-Blaisdell planer with patent dates from the 1860s, a big Tee-type faceplate lathe - all lineshaft-driven machinery. I got the nameplates (old-time script, in cast bronze) as they were picking the machinery up with an electromagnet on a crane and dropping one piece on the others to reduce it for the scrap heap. I was living in a little apartment while working as an engineer on a power-plant construction site, so could not "save" the Flather lathe.
We are now running the 60" swing x 24' LeBlond lathe, turning wicket gates from some of our hydro turbines. This is akin to turning ship's rudder paddles; we add about 500 lbs of static balance weights to these wicket gates before they go into the lathe and we get a real smooth job of it. The gates did not have centres in the bottom journals for support by tailstock so I came up with a scheme - and we put the centres in using our planer mill. That is a Cincinnati Hypro open-side 4' x 4' x 21' planer. It was built in 1939 for the naval Shipyard at Norfolk, Virginia. In 1947 it was sold to a shop in Philadelphia, PA. They added two 30 HP milling heads in 1976 and put digital readout on in 1980s. We had the machine re-scraped and added state-of-the-art "Vector drives" (stepless wide range AC motor drives) to all the feeds and installed a right-angle head on the vertical milling spindle. We set the wickets up on large vee blocks and indicate the journal to centre of the milling spindle and then put our centre in using the milling spindle - we wind up with a centre almost 3" diameter ! I have been meaning to send you some pictures …..