A lot of railways disappeared before 1940, but some disappeared much earlier than that and most of the time without leaving a trace. Looking at recent topographical maps one can see lots of stretches of former railways, but that’s most of the time a recent, standard gauge line through relatively flat country, where the railway embankment can be easily seen in the landscape and so will often be included on maps.
But it is rare to discover a railroad which from the start was pretty much a failed project and consequently was dismantled very early on. The Nantucket Railroad, opened for traffic in July 1881, was originally proposed to be part of the development of Nantucket Island as a holiday resort, to combat the rising economical problems connected with the whaling industry which was on its last legs on Nantucket by that time. However, money was short and so only in 1884 did the line reach its full length, 11 miles. The original plan envisaged a line of 17 miles, which was remarkable, because the distance as the crow flies between the two towns connected was actually 7 miles…
The company went pretty much bankrupt in 1894 for the first but not the last time. The line managed to hold on until 1918 in the end, but only because until then automobiles were restricted on the island and other shorter routes were planned but never developed. And so the line ended its short and unsuccessful life.
The line ran from Nantucket in the middle part of the island to Siasconset on the eastern seaboard. That meant the line ran partially next to the sea and in 1893 the sea caused bankruptcy of the company by wiping away part of the line near Tom Never’s Head. And that is approximately where the following screenshot was taking. This is a USGS map showing an old railroad bed running almost straight through the island towards the sea next to Siasconset.
More information on the railroad can be found on Wikipedia at http://en.wikipedia.org/wiki/Nantucket_Railroad. There some details are different from the ones I mention here, my details come from the wonderful book “American Narrow Gauge Railroads” by George Hilton. The book has a picture of engine number 2, the Wikipedia page also has a picture (but a different one).
There is also something else: a documentary! It’s quite interesting, with a lot of material in it. See it here at http://www.joost.com/120000n/t/Another-Island-Story-Nantucket-Railroad#id=120000n.