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In the UK, only from 1845, when legislation and land costs began to constrain the laying out of rail routes and tighter curves were necessary, were the principles beginning to be applied in practice.
[[File:Brusio Viaduct (158241421).jpeg|thumb|[[Brusio
The 'true spiral', whose curvature is exactly linear in arclength, requires more sophisticated mathematics (in particular, the ability to integrate its [[intrinsic equation]]) to compute than the proposals that were cited by Rankine. Several late-19th century civil engineers seem to have derived the equation for this curve independently (all unaware of the original characterization of the curve by [[Leonhard Euler]] in 1744). Charles Crandall<ref>{{cite book
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