He was born at Knock, near Balrothery in County Dublin (now Fingal), on 26 March 1729, the eldest son of Alexander (died 1768)[1][2] and Isabella Hamilton.

Hamilton entered Trinity College Dublin on 17 November 1742 at the age of 13 with Thomas McDonnell as his tutor.

[3] He took the competitive examination for a vacant fellowship of the college in 1750, but the position was secured instead by his friend Richard Murray, who was a few years older.

[5] He was appointed Erasmus Smith's Professor of Natural and Experimental Philosophy at Trinity College Dublin in 1759[6] and that same year graduated Bachelor of Divinity (BD).

[9] While dean he also acted as treasurer for the infirmary or county hospital, he established Sunday schools in the districts of the parish, and he founded a charitable loan for poor tradesmen.

Hamilton wrote a mathematical treatise on conic sections called De Sectionibus Conicis: Tractatus Geometricus, published in 1758.

[17] It was "soon adopted in all the British universities"[13] and was translated from Latin into English as A Geometrical Treatise of the Conic Sections in 1773.

[3] Hamilton married Isabella, daughter of Hans Widman Wood of Rosmead, County Westmeath, in 1772.