He was born at Peakirk near Peterborough in Northamptonshire, and died at Milton in the same county. He lived a very retired life, and saw little or nothing of society; when he did mingle in it, his dogmatism and pugnacity caused him to be generally shunned.
In 1762 he was appointed agent to the Earl Fitzwilliam, and held that office to within two years of his death. He was first known as a mathematician by his essays in The Ladies' Diary for 1744. In 1766 he was elected a fellow of the Royal Society.
He was well acquainted with the works of the mathematicians of his own time, and has been called the English d'Alembert. In his Discourse on the Residual Analysis, he proposes to avoid the metaphysical difficulties of the Method of Fluxions by a purely algebraic method. The idea may be compared with that of Joseph Louis Lagrange's Calcul des Fonctions.
His memoir (1775) on the rotatory motion of a body contains (as the author was aware) conclusions at variance with those arrived at by Jean le Rond d'Alembert and Leonhard Euler in their researches on the same subject. He reproduces and further develops and defends his own views in his Mathematical Memoirs, and in his paper in the Philosophical Transactions for 1785. But Landen's capital discovery is that of the Landen's transformation (obtained in its complete form in the memoir of 1775, and reproduced in the first volume of the Mathematical Memoirs) for the expression of the arc of an hyperbola in terms of two elliptic arcs. His researches on elliptic functions are of considerable elegance, but their great merit lies in the stimulating effect which they had on later mathematicians. He also showed that the roots of a cubic equation can be derived by means of the infinitesimal calculus.
- The Ladies' Diary, various communications (1744-1760)
- papers in the Phil. Trans. (1754, 1760, 1768, 1771, 1775, 1777, 1785)
- Mathematical Lucubrations (1755)
- A Discourse concerning the Residual Analysis (1758)
- The Residual Analysis, book i. (1764)
- Animadversions on Dr Stewarts Method of computing the Sun's Distance from the Earth (1771)
- Mathematical Memoirs (1780, 1789)
- This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). Encyclopædia Britannica (11th ed.). Cambridge University Press
- O'Connor, John J.; Robertson, Edmund F., "John Landen", MacTutor History of Mathematics archive, University of St Andrews.
- Mathematical Lucubrations (Google Books)
- Mathematical Memoirs (1780, Google Books)