Finlay was educated at Liverpool College, completed a BA with honours at Trinity College, Cambridge in 1873 and then a Master's degree also from Trinity College. Although he was not an astronomer, on the strength of his mathematical ability he was appointed first assistant at the Royal Observatory, Cape of Good Hope in April 1873, under the direction of Edward James Stone. His first duties were the reduction of the meridian observations of the previous director, Thomas Maclear. In November of 1873 he was elected a fellow of the Royal Astronomical Society. From initially being inexperienced in practical astronomy he went on to become an expert in the use of the Heliometer. His chief interest was comets and in March 1880 he presented a paper on them before the South African Philosophical Society. He wrote more than 10 papers on the observations and orbits of comets. He was the first astronomer to record observations of the "Great Comet of 1882", and he presented his findings to the South African Philosophical Society in October 1882. On 6 December 1882, Finlay and Robert T. Pett, the third assistant at the Royal Observatory, observed the transit of Venus from a temporary observatory they had erected at the instruction of David Gill. In September 1886, Finlay discovered another comet and computed its orbit. This comet is named Comet Finlay after him. His other astronomical work included the preparation of star correction tables and determination of Stellar parallaxes, in collaboration with David Gill. In 1887 he undertook an analysis of the tidal records of Table Bay and Algoa Bay. He also was involved in the determination of the relative longitudes of Cape Town, Aden, Delagoa Bay, Quelimane and Zanzibar. He also participated in the measurement of a baseline near Kimberley, Northern Cape in 1891. From 1880 to 1889 Finlay acted as an examiner for mathematics and physics at the University of the Cape of Good Hope. In 1886 UCGH awarded him a Master's degree in mathematics. Finlay became chief assistant at the Royal Observatory in 1897 but was forced to retire in August 1898 owing to poor health. He went to England to recuperate but returned to South Africa. He left retirement and stood in as professor of mathematics at the South African College in 1909, on behalf of T.P Kent. In 1914 he replaced David Williams at Rhodes University College and taught mathematics and surveying. From 1913 to 1915 Finlay acted as examiner in mathematics at the University of the Cape of Good Hope. He continued to work up until his death in 1923.
