After taking the baccalaureate in 1849, he studied mathematics in Bourges, near where his family lived, and after continuing his studies in Paris entered the École Polytechnique in 1853.
He served as a lieutenant in the French artillery in the Second Italian War of Independence, in 1859, and became a captain in 1863.
His military decorations include the cross and officer's rosette of the Legion of Honour, awarded in 1868 and 1886 respectively.Beginning in 1879, Delannoy began a correspondence with Édouard Lucas on the subject of recreational mathematics and probability theory; he eventually published eleven mathematics articles.