The two photos that triker has posted show one track positioned a few feet southerly from Dedham Square Depot that stood about where the Providence Highway intersects with Eastern Ave. at the base of the municipal parking lot now. It has been described as "an impressive stone edifice". The tracks pictured would have been one of two almost parallel lines coming from Islington to Dedham Square, one originally built by the Boston & New York Central RR (later absorbed by the NYNE RR) and the other built by the New York and New England Railway in 1849 and 1890, respectively. The double restraining rail on a straight section indicates that it was used for protection, probably due to it's proximity to the Depot.The Row's of both almost entirely sit underneath the Providence Highway now. The tracks built by the NYNE RR Dedham Branch were in direct competition with the Boston and Providence's Dedham Branch to and from Boston. The largely unused originally B&NYC RR ROW (probably that in the pictures) were used, as Leo stated, for storage, while the almost parallel NYNE RR line was used for passenger routing through Dedham Square. (I think that those are remarkable photos as I am not aware of anywhere now where uncovered street running rail exists, abandoned circa 1883 !) Boston and Providence Railroad tracks built in 1850 came through between what is now the VFW Parkway and Crosstown Ave. in West Roxbury, running to the rear of Toys R Us as they crossed over into Dedham. The tracks then kept going southerly, right through what is now the Stop N' Shop off Providence Highway in Dedham, right through the present Dedham Mall, continuing to a B&P RR yard with a car shed, depot, turntable, and engine house. The car shed was just about where the overpass is now, and there was a spur that left the line, accessing the turntable to the five-stall engine house where the municipal parking lot is currently accross from the CVS on Eastern Ave. Coming east/west out of what is the field at East/Avery Sts. now, was a B&P RR ROW that crossed the north/south Dedham Branch, switching into the depot or entering these yards to alongside the engine house. This ROW, built in 1834 by the B&P RR was also part of the B&P RR Dedham Branch and was the first branch line in New England. If you counted all the railroad tracks around the vicinity of Dedham Square (not including street railway), including all the tracks on the Dedham Branches of both the B&P RR and the NYNE RR (2), Depot (4 incl. Dedham Branches), Avery St. (1), and Turntable and Enginehouse (5), there were 10. If you look carefully, there is still some evidence left.
Link to Ward Maps for 1876 Dedham. Magnify and swing over to end of Bryant St. for detail of Depot, Yard, etc.
(On a side note, it was this Dedham Square Depot that the infamous D.B. Torrey, the Boston and Providence Railroad's 4-4-0 involved in the Bussey Bridge Disaster, left from prior to it's fateful encounter with the bridge in Roslindale that collapsed, killing 23 passengers on the snowy morning of March 14, 1887. The Bussey Bridge Disaster is considered the first major railroad disaster in America, and is credited with the population of then countrified Roslindale.)