American, Sylvester Howard Roper (1823-1896) invented a two-cylinder, steam-engine motorcycle (powered by coal) in 1867.  This can be considered the first motorcycle, if you allow your description of a motorcycle to include a steam engine. Howard Roper also invented a steam engine car.

    German, Gottlieb Daimler invented the first gas-engined motorcycle in 1885, which was an engine attached to a wooden bike. That marked the moment in history when the dual development of a viable gas-powered engine and the modern bicycle collided. Gottlieb Daimler used a new engine invented by engineer, Nicolaus August Otto. Otto invented the first "Four-Stroke Internal-Combustion Engine" in 1876. He called it the "Otto Cycle Engine" As soon as he completed his engine, Daimler (a former Otto employee) built it into a motorcycle