Edward came to England in 1842 and his father worked, still as commissary-general, in London. Edward attended University College School and then entered University College, London, in 1847 having won a scholarship. There he studied under De Morgan whose influence led to him deciding on a career in mathematics. After the award of a B.A. from London in 1849, he entered Peterhouse on 1 June 1850 at the same time as Maxwell. However, Maxwell transferred to Trinity College (perhaps because he felt Routh was too strong competition!). Routh obtained his M.A. from London in 1853, being awarded at that time the Gold Medals for Mathematics and for Natural Philosophy.
In January 1854 Routh graduated with a B.A. from Cambridge. He was Senior Wrangler in the Mathematical Tripos examinations (ranked first among those with First Class degrees) with Maxwell being placed second. The prestigious Smith Prize was at that time decided by examination, and the prize was divided equally between them (the first time the prize had been awarded jointly). In 1855 Routh was elected a fellow of Peterhouse and was appointed as a College lecturer in Mathematics. In the following year he was appointed as assistant tutor at Peterhouse.
In 1857 the post of first assistant at the Royal Greenwich Observatory became vacant and Routh was invited by Airy, the Astronomer Royal, to visit the Observatory so that he might be offered the post. He did go to the Observatory but decided that he would prefer not to accept the post there, but rather remain at Cambridge. While he was there, however, he met Hilda Airy, George Airy's eldest daughter, and a friendship began which led to their marriage on 31 August 1864. They had five sons and one daughter. Edward Airy Routh served as a lieutenant in the royal artillery, George Richard Randolph Routh became an inspector of schools, Arthur Lionel Routh served as a lieutenant in the royal artillery, Harold Victor Routh became professor of Latin in Toronto, and Rupert John Routh served in the Indian civil service.
Routh became the most famous of the Cambridge coaches for the Mathematical Tripos. His first pupil was Third Wrangler in 1856 and, two years later, both the First and Second Wranglers were his pupils. By 1862 he was established as the best Cambridge coach for, in that year, he coached 19 of the 32 Wranglers including seven of those placed in the top ten. Of course once his reputation was established the best students sought him out as a coach and so maintaining his leading role became relatively easy. Naturally his exceptional teaching ability was a factor in his success, but equally his understanding of where students should allocate their energies and how they could make the best use of their knowledge were important. Over a period of 22 years from 1862 Routh coached the Senior Wrangler in every year. During his career he coached about 700 pupils of whom about 480 were Wranglers out of around 900 Wranglers over these 30 years.
When he retired as a coach in 1888 a presentation was held at which Routh's portrait, painted by Sir Hubert von Herkomer, was presented to his wife. Eighty of his former pupils had contributed to the cost of the painting. In fact a rather amusing story was told at the time of the presentation to illustrate Routh's skill as a teacher :-
The case of a student of hydrodynamics was alleged as typical of the trials to which [his patience] was exposed. The troubled undergraduate's primary difficulty lay in conceiving how anything could float. This was so completely removed by Dr Routh's lucid explanation that he went away sorely perplexed as to how anything could sink!We must not think of Routh as just a superb teacher, however, for he also contributed to mathematics with some excellent research papers and some outstanding texts. The research areas which interested him most were geometry, dynamics, astronomy, waves, vibrations and harmonic analysis. His work on mechanics was particularly important and in 1877 he was awarded the Adams Prize for work on dynamic stability Treatise on the stability of a given state of motion, particularly steady motion. The fact that he did this in a Christmas vacation suggests that had he devoted more time to research and less to teaching he may have had a much more lasting impact of the course of mathematics. In fact the impact of this prize winning work was very significant since Thomson and Tait rewrote for the second edition of their text Natural philosophy treatise the part dealing with equations of motion using Routh's developments.
He published famous advanced treatises which became standard applied mathematics texts such as A Treatise on Dynamics of Rigid Bodies (1860), A Treatise on Analytic Statistics (1891), and A Treatise on Dynamics of a Particle (1898).
He was elected a fellow of the Cambridge Philosophical Society in 1854, and in 1856 he became a founder member of the London Mathematical Society. He was also elected a fellow of the Royal Astronomical Society in 1866 and of the Royal Society in 1872. He was awarded honorary degrees from a number of universities including Glasgow (1878) and Dublin (1892). He was made an honorary fellow of Peterhouse in 1883.
The author of  writes:-
Dr Routh was a man of the most kindly disposition, and was both liked and respected by his numerous pupils. He influenced deeply the mathematical teaching of his time and held strong views as to the best ways of promoting the study. For years he has been a familiar figure in the roads and paths around the University town, but latterly his health failed and he was unable to take his usual walk. The alterations in the procedure of the Mathematical Tripos adopted by the Senate last autumn were a real grief to him, and almost his last appearance in public was at the debate on the proposed changes, when he fought for the retention of the Senior Wrangler.
Article by: J J O'Connor and E F Robertson