He was born near Manchester[1] and educated at King's School in Chester, before winning a scholarship to study at Sidney Sussex College, Cambridge, in 1916.
[2][3] However, he decided to postpone Cambridge, and instead enlisted for service in the First World War, finding an interesting role in the Royal Aircraft Establishment in Farnborough, to serve doing applied aeronautical science alongside George Paget Thomson.
[1] Returning to Cambridge after the war he graduated MA and then continued as a postgraduate, receiving a PhD from Imperial College London in 1924.
Whilst Professor of Natural Philosophy at Aberdeen University, Carroll had been interested in acquiring desk computing machines for his students.
These greatly reduced the labour in producing the mathematical tables needed in astronomy and other fields, including gunnery.