The subject bicycle is what is generically referred to as spur gear drive. In mid-1890s the bicycle industry experienced a boom. resulting in a plethora of new manufacturers. The increased competition led to increasingly lower prices and profits and by the very late 1890s market saturation resulted in an industry recession. At the time, the chain was viewed as the primary area for bicycle improvement and many manufacturers focused on developing chainless models, as a means to increase their market share.

The most common solution was shaft drive but spur gear drive was one alternative selected by some manufacturers. The example in the Museum of Science and Technology is a Caroll (see attached), who patented the bicycle spur gear drive in the USA in 1895 but were out of business before the turn of the century. Personally, I've seen about a handlful of surviving Carroll and there are other known manufacturers of spur gear drive bicycles including Humber and Hildick, so it's not a unicorn, though they are still very rare. The issues with spur gear gear drive were the extra weight and cost, which prevented it from becoming popular. It was primarily a novelty and had largely disappeared by the turn of the century, though there have been periodic attempts at resurrection.