All this modern day high tech stuff is amazing....
... thing is, they started building the London Underground back in the 1850's with Men with Picks, Axes and Donkeys to remove the rubble, mud and clay so that the first Tube Train could run in 1863 without fault, not sure they had all this high tech flangle dangle stuff back then let alone how to work out the curvature of the Earth.
The first underground tunnel under the Thames, and indeed the first one under any river in the World, was started in 1825 of a design by March Brunel. His son, Isambard, became the director of operations and, after a number of floodings, tunnel collapses, with almost the loss of Isambard's life during one, it was completed in 1843. During this construction, for the very first time, a moving 'shield" was used with men on different levels on it cutting away at the London mud and rock. This was the forerunner of the modern mechanical shields, or cutting machines, as used to construct the Crossrail tunnels.
Brunel's Thames Tunnel, after proving to be a very popular walkway under the river from one side to the other, was sold to a railway company and in 1865 and used by trains as part of the new London Underground system.
As for accounting for the curvature of the earth being taken into account, for all us sane people who do not believe the Earth is flat, it is purely common sense that this is so. Someone like any of the 19th century engineers, first building the canals, then the railways, when surveying across vast lengths of countryside would have found it very obvious that in their sights the landscape was bending down and thus requiring allowance in their calculations when building tunnels. However, when the Foulridge Tunnel was built on the Leeds to Liverpool canal between 1796 and 1801, the engineers did have problems lining up the bore of the tunnel to where they wanted to go. This must be understandable though, as zirk implies, methods of construction was crude and without any of the modern technical devices of today!