Andræ was a son of the Prime Minister. He graduated in law in 1868 and became a civil servant in the colonial and finance administration. In 1894 he obtained an early discharge from the civil service, with a pension, by producing a medical certificate issued by the Austrian psychiatrist Richard von Kraft-Ebing, for "nervousnes" caused by "contrary sexuality".

He later wrote a very long and learned article under the pseudonym "Tandem" (Latin: "at last"), titled The Feeling of Contrary Sexuality in the leading journal of medicine. Tandem's account of the experiences of a "patient who cannot avoid ejaculations" while watching naked young men on a nearby bathing raft was an exemple of the abnormally early and strong sexual drive of the "contrary sexual", but may also have been a personal memoir.

The author demonstrated that "contrary sexuality" was not opposed to respectability and could be combined with learning, insight and intelligence, and found in society's better classes. Thus Andræ was the first in Denmark to publicly defend homosexuals.