It is unclear exactly why a separate company was established to produce the Rapid. Tyler Anderson, author of the definitive book on the Fox brand, aptly named The Fox Typewriter Company (2016), speculates that it was to sell Fox No.23 overstock inventory. It is a very plausible theory. Rapids were advertised in 1912 and sold for $75. A Rapid No.24 model was later introduced but, like its predecessor, it too was nothing more than a rebranded Fox. Serial numbers for the Rapids ranged in the 35k's.
The practice of using up old stock to produce a new-ish product neither starter nor stopped with the Rapid. Ten years later, in 1922, the Annell' followed a similar business model.
View an original Rapid Typewriter Co. letterhead here...
Rapid Typewriter Co.
Grand Rapids, Michigan, US
Not to be confused with the other, more rare Rapid, this traditional looking typewriter, the Rapid No.10 model, was nothing more than a rebranded version of the Fox No.23. And though it may appear to have been produced by different companies, they were not. The Rapid Typewriter and the Fox Typewriter Companies were actually one and the same.
The two ads on this page were both published in 1912 and both mentioned a "1912 Rapid Calendar and Catalog." I would be thrilled to find either of these two pieces of ephemera, or any other.