Both were the first mountain cogwheel railways in their respective continents.

Mount Washington Cog Railway in 1868, first in the USA.
Mount Rigi Cogwheel Railway in Switzerland, in 1871 the first in continental Europe.
Both lines are still running today.

The first mountain cog railway was the Mount Washington Cog Railway in the U.S. state of New Hampshire, which carried its first fare-paying passengers in 1868. The track was completed to reach the summit of Mount Washington in 1869.

The first mountain rack railway in continental Europe was the Vitznau-Rigi-Bahn on Mount Rigi in Switzerland, which opened in 1871.

The definition of a rack railway (also rack-and-pinion railway, cog railway, or cogwheel railway) is a steep grade railway with a toothed rack rail, usually between the running rails.

Trains are fitted with one or more cog wheels or pinions that mesh with this rack rail. This allows the trains to operate on steep grades above around 7 to 10%, which is the maximum for friction-based rail.

If you looked underneath the train, this is what a rack and pinion looks like.

The rack lies between the rails. The pinion gear is connected to actual train car.

As of 2009, the Mt. Rigi locomotive is still used in historic rides. Built in 1858 (the plate on the loco and other sources show the later date, 1873), it is the oldest Swiss loco surviving and also the only remaining standard gauge rack fitted vertical boilered loco in the world.

The two cylinder steam engine provides 196 PS power under 7.5 km/h speed. The loco was removed from service in 1937, when the railway was electrified, but has been returned in 2009.

The Mount Rigi cogwheel is used during many of the ECHO Rails & Trails tours. There is a Day Mount Rigi tour that includes some hiking and Swiss sausage grilled lunch outside and also a Day tour that includes both Mt. Rigi & Mt. Pilatus. Both mountains are included in our 12 Day Grand Swiss Rail tour.