This included his formulation of the method of least squares while working on a surveying problem (in two proofs of the exponential law of error published independently of Carl Friedrich Gauss) for which he is chiefly remembered.

His father, of French Huguenot descent, was a school teacher and maker of mathematical instruments, and he apparently received a good education until he was fifteen when both his parents died.

[3][4] In the cause of democratic reform and national independence, on 7 June 1798 he led a contingent of United Irishmen in the rebel army commanded by Henry Joy McCracken at the Battle of Antrim.

While he failed to attract sufficient subscribers,[6] the first volume of the Analyst has been considered "the best collection of mathematical work produced in the United States up to that time".

[4] In 1834, Adrain was asked to resign from the University of Pennsylvania on grounds of class ill-discipline (instances of students overturning benches and throwing eggs).

Among his broad interests in physics, astronomy and geography, his paramount concern was dynamic geodesy, specifically the mathematical investigation of the shape of the earth.