Jointly, with fellow engineers Olivier Zschokke and Adolf Naef, he submitted an application to the Canton of Luzern (Lucerne) for permission to build his line. The Canton administration already knew of the Mount Washington Railway in the United States using a similar system devised by Sylvester Marsh (and known as the Marsh system in the USA) and saw the advantages in this construction, granting permission on 9th June 1869. The construction itself began in the following September, the limited liability company, which had offered 1250 shares was greatly over-subscribed on the first day of issue. On 21st May 1870, Riggenbach’s birthday, locomotive No.1, named Stadt Luzern, made its first trial run. Exactly one year later the first mountain railway using rack and pinion technology was officially opened, Riggenbach, never noted for missing an opportunity, drove the first train to the upper terminus at Rigi Staffelhohe, the boundary of Cantons Luzern and Schwyz. As no application for the line had been submitted to the Canton Schwyz authorities this was the limit of operation.
The line, from Vitznau to Rigi Staffelhohe was 5 km long and climbed a total of 1115 m. to reach a height of 1550 m. (3456.5 ft) at its summit, the maximum gradient being 1 in 4 (25%).
The concession to construct the Arth - Rigi Railway was ceded, in 1873, to the International Company for Mountain Railways in Aarau, a company founded by Riggenbach and this company carried out the railway project as general contractor at a cost of CHF 4.2 million and also supplied five of the six steam locomotives needed to operate it. Construction of the first section, that from Arth-am-See, a station by Lake Zug, to Oberarth commenced in 1873 and once the position of the railway station in Arth (now known as Arth-Goldau) had been agreed with the main line company, in 1874, construction work on the second section began, the Arth - Rigi Railway becoming operational on 4th June 1875. The line offered only summertime services until 1884 when year-round operation commenced.
The station at Arth-Goldau is situated above the tracks and platforms of the SBB-CFF-FFS and was opened in 1897 becoming the lower terminus of the railway following the closure of the line to Arth-am-See.
The first section, that from Rigi Staffelhohe to Rigi Kulm, was just 1.8 km. long but enabled trains to reach the summit at Rigi Kulm, a height of 1752 m. above sea level, a climb of a further 202 m. (606.2 ft) from Rigi Staffelhohe. The line from Arth to the junction at Rigi Staffelhohe was 6.8 km. long, making the total length of the Arth line 8.6 km. The maximum gradient is 1 in 5 (20%)
A new rail connection was built at Rigi Staffel, linking the Arth-Rigi Railway and the Vitznau-Rigi Railway, and opened in 1990. This connection involves a single line leaving the Vitznau line below the station and running parallel to its own line until it joins passing loop line from Goldau. This enables trains from Vitznau to arrive at and depart from Rigi Kulm in the platform normally used by the Arth-Rigi trains. Arth-Rigi trains can access the Vitznau line by the same means.
The construction period was short, a mere eleven months saw the new aerial cableway completed and the opening took place on 15th July 1968. The cable car rises from the lakeside some 924 m. (2865 ft) to its summit and has a journey time of just 10 minutes. A 100-metre pathway at Rigi Kaltbad connects the mountain station of the cable car with the cogwheel railway station. The technical installations were the work of K. Garaventa & Sons of Goldau, and the large cabins were supplied by the Carrosseriewerke Company of Aarburg. On the 25th anniversary of the opening of the cable car (in 1993), the original two red passenger cabins were replaced by modern panorama cabins.
In 1992 the two railway companies merged to become the Rigi Railways Company and in the same year took over the Rigi Ski Lift Company.
Services from Vitznau operate more seasonal than those from Arth with more trains in the summer months than in Spring, Autumn and Winter. In Winter the service is approximately hourly with early and late trains terminating at Rigi Staffelhohe, only Vitznau depatures between 09h00 and 17h00 continuing to Rigi Kulm. In the Spring and Autumn all services from Vitznau with the exception of the first and last trains complete the journey to Rigi Kulm and in the Summer period (from the end of May - British Spring Bank Holiday weekend, to the end of September) the service is increased to 15 trains per day with, again, only the first and last not completing the journey to the summit. Journey times are 30 minutes (up) and 40 minutes (down). The steam service operates Sundays from July to September with a single round trip and, like is counterpart from Arth, the fare carries a CHF 10.00 supplement.
The Weggis - Rigi Kaltbad cable car operates year round, except for two short (timetabled) maintenance periods, usually the first week in April, and from mid-November to the end of the month. The service is half-hourly and the journey takes 10 minutes.
The line operates two steam locomotives, No's 16,(SLM No.2871, built 1923) (See photograph above) and 17,(SLM No.3043, built 1925). They are painted in a dark green livery, lined red. They also have three electric locomotives, a Stadler built battery operated class Ta2/2 in red livery, which carries the identification VRB 1, and usually to be found shunting at Vitznau, a class He2/3, built in 1930 by SLM / MFO, again in red and numbered 8, which is the Arth snowplough locomotive, and a further class He2/2, in red livery and numbered 18, built by SLM in 1938 and which is usually based at Vitznau.
The railcars, class Bhe2/4, are painted in a red livery for operation on the Vitznau section. Numbered 1 - 4 (inc) they were products of SLM / BBC, and introduced with the electrification of the line in 1937. To add to the passenger carrying capabilities of the line a class BDhe4/4 from the same makers arrived in 1964 and, again from SLM / BBC, in 1986 two railcars (BDhe2/4), numbered 21 and 22, arrived with single end driving trailers (Bt) numbered 31 and 32. Although the railcars are capable of working as single units they are normally to be found working with the trailers.
On the Arth section class BDhe2/4 railcars, built by SLM / SAAS and numbered 11 and 12 arrived in 1949, being joined by No.13 in 1954 and No.14 in 1967. These were joined by class BDhe4/4 No.15 in 1982. Class Bt driving trailers followed the railcars, No.21 and 22 in 1958, No.23 in 1960 and Nos. 24 and 25 with the railcars in 1967 and 1982 respectively. These vehicles are liveried in blue/white. It is common, on light traffic days, for the railcars to operate without trailers.
Unusually for these days, the drivers control cab, in some older railcars, is not separated from the passenger section.
The line also has a collection of historic coaches and those for special use. The most usual use for these vehicles is with the steam locomotive, where the train usually consists of three coaches.
Worthy of a special mention is Class BDhe 2/3, Railcar No.6, the world's first cogwheel-railcar which dates from 1911. Working with yellow liveried coach, class B2, No.35, built in 1899, this forms the Rigi Pullman train. There are upholstered seats for passengers who like comfort, wooden benches for the more hardy and a standing bar for those who can remain steady on their feet. The bar is staffed by hostesses dressed in outfits dating from its construction.
The line is also home to another early example in Railcar No.7, a 1925 built BDhe2/4 built by SIG / SLM / MFO. This works from Arth, frequently with a coach, or in the wintertime, the toboggan wagon.