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History of Mathematics in Russia Before 1917 cover
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History of Mathematics in Russia Before 1917

by A. P. Yushkevich

Translated by Valery Manokhin, PhD

A. P. Yushkevich’s history of mathematics in Russia from its beginnings to 1917, in first faithful English translation from the original Russian edition. Four parts and 26 chapters: numeration, metrology and the arithmetic and geometric manuscripts of old Rus’; Magnitsky’s Arithmetic and the schools of Peter the First; the Petersburg Academy of Sciences, with five chapters on Euler — infinite series, his trilogy on analysis, differential equations and the calculus of variations — and one on his students and successors, Fuss among them; Lobachevsky and the discovery of non-Euclidean geometry; Ostrogradsky and Bunyakovsky; Chebyshev and the Petersburg school, with chapters on number theory, probability theory, the approximation of functions, differential equations, series, analytic functions, geometry and algebra; and the rise of the Moscow school of function theory under Egorov and Luzin. The school textbooks of Kiselev, Davidov, and Malinin and Burenin appear in their place. Fifty numbered figures with portraits and facsimiles, and a name index. 673 pages.

26 chapters · 673 pages

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Where this fits in school

Full curriculum table — every book by school year →
United StatesUniversity and adultHistory of mathematics — a reading text, not a course
United KingdomUndergraduate and adultHistory of mathematics; no qualification maps to it
CanadaUniversity and adulthistory-of-mathematics reading

This book matches no modern course. It is worth reading for the mathematics; it will not put a mark on an exam paper.

Not covered in this book

exercises — this volume is exposition only

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Table of contents

250 section headings — the printed contents page in full.

Translator's Preface

Author's Preface

Part One. Mathematical Culture in Russia before the Beginning of the Eighteenth Century

Chapter 1. Mathematical Knowledge in Ancient Rus'7 sections
  1. Old Russian numeration
  2. Metrology
  3. The first systems of fractions
  4. Mathematical monuments of Kievan Rus'
  5. Kirik of Novgorod
  6. Questions of the “philosophy of mathematics”
  7. The Tatar yoke
Chapter 2. Manuscripts of the Fifteenth–Seventeenth Centuries14 sections
  1. The rise of the Muscovite state
  2. Arithmetic in seventeenth-century manuscripts
  3. Instrumental reckoning
  4. The rule of three
  5. The rule of false position
  6. Problems for amusement
  7. Geometric progressions
  8. The problem “of knowing the money in a heap”
  9. Terminology
  10. An assessment of the arithmetic manuscripts
  11. The measurement of figures
  12. The geometric manuscript “Synodal No. 42”
  13. Problems of the continuous and of indivisibles
  14. Summary
Chapter 3. The State and Mathematics in the Era of Peter the First8 sections
  1. Training of military and technical personnel
  2. New schools
  3. Ya. V. Bruce
  4. L. F. Magnitsky
  5. A. D. Farquharson
  6. Magnitsky's Arithmetic
  7. Algebra and trigonometry in Magnitsky
  8. The first textbooks of geometry and trigonometry

Part Two. Mathematics in the Petersburg Academy of Sciences in the Eighteenth Century

Chapter 4. The Petersburg Academy of Sciences4 sections
  1. The founding of the Academy of Sciences
  2. Mathematics in the Academy of Sciences
  3. The Academy of Sciences and mathematical education
  4. Educational literature
Chapter 5. The First Academician-Mathematicians16 sections
  1. Mathematics at the turn of the seventeenth and eighteenth centuries
  2. The first academicians
  3. J. Hermann
  4. Differential equations
  5. Questions of geometry
  6. F. C. Maier and the development of trigonometry
  7. N. Bernoulli
  8. G. W. Krafft
  9. C. Goldbach and the theory of series
  10. Goldbach's problems in number theory
  11. Integration of the differential binomial
  12. Daniel Bernoulli
  13. Problems of the theory of oscillations
  14. Numerical solution of equations
  15. Probability theory and statistics
  16. The error distribution curve
Chapter 6. Leonhard Euler5 sections
  1. Beginning of his career
  2. Move to Petersburg
  3. Euler in Berlin
  4. Euler's return to Petersburg
  5. General characterization of his work
Chapter 7. Infinite Series4 sections
  1. Interpolation of sequences and series; special functions
  2. Euler's summation formula
  3. The zeta function
  4. Summation of divergent series
Chapter 8. L. Euler's Mathematical Trilogy17 sections
  1. Introduction to the Analysis of the Infinite
  2. The concept of function
  3. Investigation of elementary functions
  4. Elements of the theory of functions of a complex variable
  5. The fundamental theorem of algebra
  6. The logarithmic function
  7. New applications of complex numbers
  8. Conformal mappings
  9. Computation of definite integrals and the d'Alembert–Euler equations
  10. The problem of interpreting complex numbers
  11. The foundations of the differential calculus
  12. Euler's “calculus of zeros”
  13. The development of the differential calculus
  14. Integral calculus
  15. Elliptic integrals
  16. Definite integrals
  17. Multiple integrals
Chapter 9. Differential Equations and the Calculus of Variations13 sections
  1. Ordinary differential equations
  2. The Riccati equation
  3. Linear equations with constant coefficients
  4. Linear equations with variable coefficients
  5. The integrating factor
  6. Singular solutions
  7. Numerical solution of differential equations
  8. Partial differential equations
  9. The problem of the vibrating string. The d'Alembert–Euler method
  10. D. Bernoulli's principle of superposition of oscillations
  11. Analytic representability of functions
  12. Trigonometric series
  13. The calculus of variations
Chapter 10. Other Mathematical Works of L. Euler5 sections
  1. Elementary geometry and trigonometry
  2. Analytic geometry
  3. Differential geometry of surfaces
  4. Topological problems
  5. Number theory
Chapter 11. Euler's Students and First Successors9 sections
  1. Euler's school
  2. S. K. Kotelnikov and S. Ya. Rumovsky
  3. A. J. Lexell
  4. N. I. Fuss
  5. F. I. Schubert
  6. S. E. Guryev's memoir
  7. Questions of the foundations of analysis at the turn of the eighteenth–nineteenth centuries
  8. S. E. Guryev and his followers
  9. Mathematics in the Academy of Sciences at the beginning of the nineteenth century

Part Three. Mathematics in Russia in the First Half of the Nineteenth Century

Chapter 12. General Conditions of the Development of Mathematics4 sections
  1. Reform of the educational system
  2. Physics and mathematics faculties
  3. The Academy of Sciences
  4. Peculiarities of the development of mathematics in Russia and abroad
Chapter 13. N. I. Lobachevsky and the Discovery of Non-Euclidean Geometry20 sections
  1. Euclid's parallel postulate
  2. N. I. Lobachevsky
  3. The worldview of N. I. Lobachevsky
  4. Space and reality
  5. Initial concepts of geometry
  6. The discovery of non-Euclidean geometry
  7. Lobachevsky on the geometry of the real world
  8. The first applications of Lobachevsky's geometry
  9. The consistency of hyperbolic geometry
  10. N. I. Lobachevsky and his contemporaries
  11. C. F. Gauss's theory of surfaces
  12. F. G. Minding's investigations in the theory of surfaces
  13. Further development of non-Euclidean geometry
  14. Geometry and physics of the twentieth century
  15. The axiomatic method
  16. Lobachevsky's Algebra
  17. The works on algebra of M. V. Ostrogradsky and O. I. Somov
  18. P. L. Chebyshev's student work
  19. Lobachevsky's investigations in the theory of Fourier series
  20. Lobachevsky's convergence test
Chapter 14. A New Upsurge of Research in the Academy of Sciences; M. V. Ostrogradsky and V. Ya. Bunyakovsky11 sections
  1. The life of M. V. Ostrogradsky
  2. Ostrogradsky's works on the theory of heat
  3. Calculus of variations
  4. Differential equations
  5. Multiple integrals
  6. Integration of algebraic functions
  7. The life and work of V. Ya. Bunyakovsky
  8. Bunyakovsky's inequality
  9. Bunyakovsky and the theory of parallels
  10. Probability theory
  11. Computing devices

Part Four. Mathematics in Russia in the Second Half of the Nineteenth and the Beginning of the Twentieth Century

Chapter 15. General Conditions and Features of the Development of Mathematics5 sections
  1. Mathematical education; the secondary school
  2. Mathematics in the universities
  3. Mathematical societies and scientific congresses
  4. Mathematical journals and other publications
  5. Principal directions of mathematical research in Russia and abroad
Chapter 16. P. L. Chebyshev and the Petersburg Mathematical School3 sections
  1. The life of P. L. Chebyshev
  2. On Chebyshev's work
  3. Chebyshev and his students
Chapter 17. Number Theory9 sections
  1. P. L. Chebyshev's investigations on the distribution of prime numbers
  2. Diophantine approximations
  3. The works of N. V. Bugaev
  4. A. N. Korkin and E. I. Zolotarev
  5. Works of A. N. Korkin and E. I. Zolotarev on the theory of quadratic forms
  6. A. A. Markov and his master's dissertation
  7. G. F. Voronoi and the geometry of numbers
  8. Theory of algebraic numbers
  9. New investigations in analytic number theory
Chapter 18. Integration of Algebraic Functions3 sections
  1. The investigations of P. L. Chebyshev
  2. O. I. Somov
  3. Chebyshev's successors
Chapter 19. Probability Theory7 sections
  1. The law of large numbers and the central limit theorem
  2. P. L. Chebyshev and the theory of probability
  3. Problems of the theory of moments
  4. Limit theorems of A. A. Markov and A. M. Lyapunov
  5. Markov chains
  6. E. E. Slutsky
  7. The axioms of probability theory of S. N. Bernstein
Chapter 20. Theory of Approximation of Functions5 sections
  1. The investigations of P. L. Chebyshev
  2. Works of E. I. Zolotarev, A. A. Markov, and V. A. Markov
  3. Interpolation and orthogonal polynomials
  4. Mechanical quadratures
  5. S. N. Bernstein and the constructive theory of functions
Chapter 21. Differential Equations26 sections
  1. Classical methods; A. Yu. Davidov
  2. V. G. Imshenetsky
  3. A. V. Letnikov and applications of “intermediate derivatives”
  4. M. E. Vashchenko-Zakharchenko and symbolic calculus
  5. New directions in the theory of differential equations
  6. S. V. Kovalevskaya
  7. Kovalevskaya's theorem
  8. The problem of the rotation of a rigid body
  9. V. A. Anisimov
  10. S. N. Bernstein and Hilbert's problems
  11. Qualitative theory of differential equations and stability theory
  12. The work of N. E. Zhukovsky
  13. The life of A. M. Lyapunov
  14. Figures of equilibrium of a rotating liquid mass and their stability
  15. Pear-shaped figures
  16. Stability of systems with a finite number of degrees of freedom
  17. P. G. Bohl and the method of fixed points
  18. Quasiperiodic functions
  19. The life of V. A. Steklov
  20. Problems of mathematical physics
  21. Theory of closedness
  22. N. M. Günter
  23. Integral equations
  24. The younger generation of the Petersburg school
  25. The methods of W. Ritz and B. G. Galerkin
  26. A. N. Krylov
Chapter 22. Convergence and Summation of Series5 sections
  1. N. V. Bugaev and the theory of conjugate series
  2. V. P. Ermakov and his convergence test
  3. Newton's parallelogram
  4. G. F. Voronoi's summation method
  5. Improvement of the convergence of Fourier series according to A. N. Krylov
Chapter 23. Theory of Analytic Functions; Special Functions12 sections
  1. The dissemination of the theory of analytic functions
  2. M. E. Vashchenko-Zakharchenko
  3. Yu. V. Sokhotsky
  4. Sokhotsky's theorem
  5. Boundary properties of analytic functions
  6. Elliptic and hyperelliptic functions
  7. B. Ya. Bukreev
  8. Other works
  9. Historian of the theory of analytic functions I. Yu. Timchenko
  10. Investigations on cylindrical functions; N. Ya. Sonin
  11. The gamma function
  12. Bernoulli functions and numbers
Chapter 24. Geometric Investigations; Questions of Mathematical Logic19 sections
  1. O. I. Somov and vector calculus
  2. Chebyshev nets
  3. K. M. Peterson and the Moscow geometric school
  4. Lectures of V. Ya. Tsinger
  5. Works on projective geometry of K. A. Andreev and A. K. Vlasov
  6. Differential-geometric investigations of B. K. Mlodzeevsky
  7. D. F. Egorov and his students
  8. Geometry in Kazan; F. M. Suvorov
  9. A. V. Vasilyev
  10. A. P. Kotelnikov and the theory of vectors in non-Euclidean spaces
  11. D. N. Zeiliger and the geometry of ruled spaces
  12. The work of D. M. Sintsov on the theory of connexes
  13. Foundations of geometry
  14. Geometry in Odessa; the axiomatics of V. F. Kagan
  15. S. O. Shatunovsky and the theory of measurement of polyhedra
  16. Axioms of the theory of magnitude
  17. P. S. Poretsky and the algebra of logic
  18. I. V. Sleshinsky
  19. The problem of the law of excluded middle
Chapter 25. Investigations in Algebra11 sections
  1. Computation of roots
  2. Solution of algebraic equations in transcendental functions
  3. Group theory
  4. Dissemination of group theory in Russia
  5. Emergence of the Kiev algebraic school
  6. D. A. Grave
  7. Students of D. A. Grave; O. Yu. Schmidt
  8. S. O. Shatunovsky and Galois theory
  9. Hypercomplex numbers
  10. The dissertation of P. E. Romer
  11. F. E. Molin
Chapter 26. Emergence of the Moscow School of Function Theory8 sections
  1. Set theory and function theory
  2. First works on function theory in Russia
  3. Theory of functions at Moscow University
  4. The theorem of D. F. Egorov
  5. N. N. Luzin
  6. The theorem on the C-property
  7. “The Integral and the Trigonometric Series”
  8. Students and followers of Luzin

Conclusion

Name Index

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Product details

Publisher
Northern Star Academic Press
Language
English
Translator
Valery Manokhin, PhD
Print length
673 pages

Published by Northern Star Academic Press · First faithful English translation from the original Russian edition

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