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Particularly associated with ports and railways in Scotland and northern England, Meik fathered two prominent engineering sons: Patrick Meik and Charles Meik. The firm they founded remains active, though it is today known as Halcrow Group Limited, taking its name from Sir William Halcrow, who joined the company in the early years of the 20th century.
After attending the University of Edinburgh, Thomas Meik was apprenticed to John Steedman, an engineer and contractor who was working in Glasgow on the Hutcheson Bridge (designed by Robert Stevenson, grandfather of author Robert Louis Stevenson).
His first long-term post was as assistant engineer to William Chadwell Mylne of the New River Company, London. In 1845, at the age of 33, Meik was appointed engineer to the River Wear Commission (responsible for maritime works around Sunderland). In 1859, the commission took over the construction of the Hendon Dock on the south side of the Wear, and Meik was responsible for the entire works (the task included a grain warehouse and a lighthouse – which, although relocated when the South Pier was shortened in 1983, still stands today). Just a few miles further north, he was also consulting engineer to Blyth Harbour from 1862.
In 1871, Meik was commissioned to design a rail freight link. The Hylton, Southwick and Monkwearmouth Railway transported coal from collieries sited along the line to the nearby port at Sunderland. The railway was subsequently acquired by the North Eastern Railway. However, later railway designs were to prove more successful for Meik. In Scotland he designed a rail link to Eyemouth, an extension to the Forfar to Brechin line, the Newburgh and North Fife Railway and the East Fife Central Railway.
Meik continued to undertake harbour commissions. His work in Sunderland gave him a reputation and he set up a practice based in Edinburgh. From here he designed the Scottish ports of Ayr, Burntisland and Bo'ness.