Each of the lives is very different from the others, although there are common threads—for example the recurrent appearance of William Ewart Gladstone and Arthur Hugh Clough. Each story is set against a specific background.
In Cardinal Manning's story, the background is the creation of the Oxford Movement and the defection of an influential group of Church of England clergy to the Catholic Church. This aspect is covered in depth to explain the movement and its main protagonists, particularly Manning's hostile relationship with John Henry Newman. Strachey is critical of Manning's underhand manipulations in attempting to prevent Newman being made a Cardinal.
The background features of Florence Nightingale's story are the machinations of the War Office, and the obtuseness of the military and politicians. Influenced by Sigmund Freud, Strachey depicts Florence Nightingale as an intense, driven woman who is both personally intolerable and admirable in her achievements.
Dr Arnold is hailed as an exemplar who established the Public School system. Strachey describes this as an education based on chapel and the classics, with a prefectorial system to maintain order. He points out that it was not Arnold who was responsible for the obsession with sport, but does make it clear that Arnold was at fault in ignoring the sciences. Although Arnold was revered at the time, in retrospect Strachey sees his approach as very damaging. Strachey also mocks Arnold's efforts at moral improvement of the general public, for example his unsuccessful weekly newspaper.
Gordon’s is the story of a maverick soldier and adventurer, whose original military achievements in China would have been forgotten. He was a mercenary who got into and out of conflicts on behalf of various dubious governments, but much of his experience was in the Sudan. The final disaster was when the Egyptian occupation of Sudan was almost completely overthrown by fundamentalist rebels, and someone was needed to retrieve the situation in Khartoum. The job fell to Gordon, whose instincts were to do anything but withdraw, and he became embroiled in a siege. The British government was put in an almost impossible dilemma, and when eventually they did send a relief expedition it arrived just two days too late. Strachey based Gordon’s story on his diaries and letters to give an account of a strong individual almost at odds with the world.

Question: What was Doctor Arnold best known for?
Answer: Establishing the public school  system.
Alternative answer: Establishing the public school system