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.23 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 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.
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 they appeared to have been produced by different companies, they were not. The Rapid Typewriter Co. and the Fox Typewriter Co. were actually one and the same.