He was born on 21 November 1903 in Johnstone, Renfrewshire the son of Thomas Gillespie a butcher and his wife, Jane Pollock.
He then won a bursary to study Mathematics and Natural Philosophy (Physics) at Glasgow University graduating MA BSc in 1924.
He did further postgraduate studies under E. W. Hobson at Cambridge University from 1924 to 1927 under a William Bryce Scholarship, gaining his doctorate (PhD) in 1932 due to a delay in submitting his thesis.
[1] He began lecturing in Mathematics at Glasgow University in 1929 under Prof Thomas Murray MacRobert and alongside Dr T S Graham.
[3] In the Second World War he served first in the Clyde River Patrol and in 1941 moved to the Air Traffic Control division of the RAF based at Prestwick.