Here's a picture of a 1904 Rambler Motor Bike. I found it at the web site for the National Motorcycle Museum in Anamosa, Iowa. It's part of their collection. That would make it a must-see for AMC Heritage fans in the area, eh, IowaEagle? There are many accounts of the Jefferys playing around with motorcycles and motor bicycles before settling down to cars, but this is the first of these that I've seen from that period...
http://www.nationalmcmuseum.org/museum/machinesII.aspHere are just a few other pictures of cycle products in the heritage from my collection:
This is a Railway Velocipede, or foot-powered transportation, from 1895, used for inspecting the tracks. Thomas B. Jeffery held a patent for one of these machines. This flyer is for an unknown brand, and there are several different types, but I suspect he would have used the bicycle type of machine:

Here's an advertisement, also from 1895, mentioning the price of the Rambler bicycle at 100 dollars, a month's pay in that time...

A year later, in 1896, Gormully and Jeffery published this slightly sarcastic ad about a jewel-encrusted Rambler bicycle which had been their display in the Columbian Exhibition and the World's Fair.

It seems the Rambler name was also used on motorbike products in Europe at least until the post-WWII period...
1949 Rambler UK Autocycle:

1953 Rambler moped from Sweden:

I don't know if these European products were related to the American Nash-Kelvinator operation or not. They both used logos that were virtually identical to the original Rambler products at the turn of the century.
And this last one is the Mitchell Motor Bicycle from 1904. The maker is shown as "Wisconsin Wheel Works" of Racine, Wisconsin. This was a company known to be related to Mitchell-Lewis Motor Company, which was purchased by Nash Motors in 1924.

There's one that I don't have any pictures of, that was also a motorcycle or motor bicycle, produced by E. R. Thomas, of Thomas Flyer fame, that was called the "Auto-Bi". I suspect that there are probably quite a few other brands of these little powered cycles out there in the family tree...
mike