The Russian edition of this book appeared in 1976 on the
hundred-and-fiftieth anniversary of the historic day of February 23,
1826, when LobaeevskiI delivered his famous lecture on his discovery of
non-Euclidean geometry. The importance of the discovery of non-Euclidean
geometry goes far beyond the limits of geometry itself. It is safe to
say that it was a turning point in the history of all mathematics. The
scientific revolution of the seventeenth century marked the transition
from "mathematics of constant magnitudes" to "mathematics of variable
magnitudes. " During the seventies of the last century there occurred
another scientific revolution. By that time mathematicians had become
familiar with the ideas of non-Euclidean geometry and the algebraic
ideas of group and field (all of which appeared at about the same time),
and the (later) ideas of set theory. This gave rise to many geometries
in addition to the Euclidean geometry previously regarded as the only
conceivable possibility, to the arithmetics and algebras of many groups
and fields in addition to the arith- metic and algebra of real and
complex numbers, and, finally, to new mathe- matical systems, i. e.,
sets furnished with various structures having no classical analogues.
Thus in the 1870's there began a new mathematical era usually called,
until the middle of the twentieth century, the era of modern mathe-
matics.