Arthur B. Welsh, son of Alexander R. Welsh and his wife Jessie, matriculated through the University of the Cape of Good Hope in 1899. During 1899-1901 he continued his studies at the South African College, Cape Town, and in the latter year was awarded the degree Bachelor of Arts (BA) in mathematics and the natural sciences by the University of the Cape of Good Hope - the only institution in southern Africa that could award degrees at the time. He was then appointed assistant to Professor P.D. Hahn* in the Department of Chemistry of the South African College, a position he held until he resigned in December 1904. Meanwhile he continued his studies as a private student and obtained the degree Master of Arts (MA) in chemistry in 1903. This was by no means the end of his outstanding performance as a student. He subsequently passed the university's Survey Certificate Examination in 1905, passed both the first and second Mining Examinations in 1906, received the Diploma in Mining Engineering in 1907, and graduated as Bachelor of Science (BSc) in Mining Engineering in 1908.
From 1902 to 1906 Welsh was responsible for meteorological observations at the second order meteorological station of the South African College. His results for these years were reported to the Meteorological Commission of the Cape of Good Hope, which published a summary of the observations in their annual Report. He was a member of the South African Association for the Advancement of Science by 1903, but his membership lapsed between 1906 and 1910.
In November 1912 Welsh was appointed assistant inspector of mines at Boksburg (later at Pretoria). By that time, or shortly thereafter, he had obtained certificates that qualified him as a mine surveyor, mine captain, and mine manager. He wrote a "Summary of report on the Murchison Range" that was included as an appendix to the Annual Report of the Government Mining Engineer for 1916. By that time he had moved to Cape Town, where he was associated with the Good Hope Tin Mines, Ltd. In 1917 he compiled a Report on Manganese deposits in the south-west districts of the Cape Province, which was published by the Department of Mines and Industries (UG34, 1917). It contained descriptions of deposits at Hout Bay, Constantia Nek, Kogel Bay, Botha's Halt, Du Toit's Kloof, Caledon and Franshoek, all in Table Mountain Sandstone. By 1922 he was assistant head of the Mines Department in the Orange Free State (now the Free State). Towards the end of his life he resided in Johannesburg.
His wife, El(i)vera de F. A. Welsh, died in 1972. They had two sons and two daughters.