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In 1892 Kagan received a degree from Kiev University, then in 1895 he was awarded a Master's Degree by St Petersburg University.
He taught at Novorossysky from 1897 until 1922, becoming a professor there in 1917. In 1922 he went to Moscow and was appointed to the chair of differential geometry there in 1934.
Kagan worked on the foundations of geometry and his first work was on Lobachevsky's geometry. In 1902 he proposed axioms and definitions very different from Hilbert. Kagan studied tensor differential geometry after going to Moscow because of an interest in relativity.
Kagan wrote a history of non-euclidean geometry and also a detailed biography of Lobachevsky. He edited Lobachevsky's complete works which appeared in five volumes between 1946 and 1951.
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