A trained surveyor, John Landen's interest in mathematics was his leisure activity. He acted as a land-agent from 1762 to 1788. He wrote Mathematical lubrications in 1755 and he sent his results on making the differential calculus into a purely algebraic theory to the Royal Society. This work appeared as Residual analysis in 1764. These ideas were taken up by Lagrange.
Landen investigated the dilogarithm in 1760, about the same time as Euler, and introduced the trilogarithm. The trilogarithm was studied further by Spence around 1809 and Kummer around 1840.
Landen wrote on dynamics, summation of series and an important transformation giving a relation between elliptic functions. This last result, known by his name, expressed a hyperbolic arc in terms of two elliptic ones. It was published in the Philosophical Transactions of the Royal Society in 1775. Applications of this result appeared in his Mathematical memoirs of 1780.
He also solved the problem of the spinning top and explained Newton's error in calculating the precession.
Landen was elected a Fellow of the Royal Society in 1766. He corrected Stewart's result on the distance of the Sun from the Earth in 1771.
In [5] his achievements are summed up as follows:-
Though foreigners gave him a high rank among English analysts, he failed to develop and combine his discoveries. He led a retired life, chiefly at Walton in Northamptonshire. Though humane and honourable, he was too dogmatic in society.