History of geometry
Classic geometry was focused in compass and straightedge constructions. Geometry was revolutionized by Euclid, who introduced mathematical rigor and the axiomatic method still in use today. His book, "The Elements" is widely considered the most influential textbook of all time, and was known to all educated people in the West until the middle of the 20th century.
Science and technology in Russia
In mathematics Nikolai Lobachevsky, a "Copernicus of Geometry", founded the non-Euclidean geometry playing important role in modern physics. In the 19th century the international recognition was also gained by such mathematicians as Mikhail Ostrogradsky and Sofia Kovalevskaya who was the first major Russian female mathematician, responsible for important original contributions to analysis, differential equations and mechanics, and the first woman appointed to a full professorship in Northern Europe. Yevgraf Fyodorov was a founder of the modern structural crystallography (Fedorov group). At least after such prominent scientist as Chebyshev the Russian mathematical school became one of the most influential ones in the world and was represented by numerous figures greatly contributing to different fields of mathematics, physics and computing sciences. Chebyshev's students included Aleksandr Lyapunov who founded the modern stability theory (lately deeply developed by such scientists as Aleksandr Andronov and Vladimir Arnold), and Andrey Markov who developed the theory of Markov chains, playing a central role in information sciences and modern applied mathematics.
In the beginning of the 20th century Nikolai Zhukovsky and Sergei Chaplygin were among founding fathers of the modern aero- and hydrodynamics and Vladimir Kotelnikov was a pioneer in information theory by independently proposing the fundamental sampling theorem. 
Andrei Kolmogorov, a leading mathematician of the 20th century, developed the foundation of the modern theory of probability and made other key contributions to broadest range of mathematical branches, such as turbulence, mathematical logic, topology, differential equations, set theory, automata theory, information theory, theory of algorithms, dynamical systems, stochastic processes, theory of integration, classical mechanics, mathematical linguistics, mathematical biology and applied sciences. Israel Gelfand is credited with many important discoveries in algebra, topology, mathematical physics and applied sciences. Sergei Sobolev developed the theory of Sobolev space which played an extremely important role in formation of modern mathematical views and introduced the notion of distributions generalizing ideas of Newton and Leibniz.