He was born in Aberdeen on 29 January 1795. He studied at the University of Aberdeen and received an M.A. degree at the age of 16, then continued studies in Medicine in London, graduating as an M.D., and was admitted to the Royal College of Surgeons of England in 1812. He then travelled to London for further studies in surgery, becoming a Member of the Royal College of Surgeons in 1812.
Jack was employed by the East India Company as a surgeon in India, where he corresponded extensively with botanist Nathaniel Wallich. In 1818 he accompanied Stamford Raffles to Sumatra where he extensively documented the rich flora of that region until his death in 1822. Much of his work, including manuscripts, drawings, and collections were destroyed by fire in 1824.
He died at Bencoolen in Sumatra on 15 September 1822.
Three plant genera are named after him: