Kagan was born in Shavli, in the Kovno Governorate of the Russian Empire (now Šiauliai, Lithuania) in 1869, to a poor Lithuanian Jewish family.
In 1894 Kagan moved to Saint Petersburg where he continued his studies with Andrey Markov and Konstantin Posse.
His students in the theory of relativity class he taught in 1921-22 included Nikolai Papaleksi, Alexander Frumkin and Igor Tamm.
Kagan worked at Moscow State University where he held the Geometry Chair from 1923 till 1952.
He wrote a definitive biography of Nikolai Lobachevsky and edited his collected works (5 volumes, 1946–1951).