Where to start
These books are a sequence, not a shelf. Answer two questions and you will land on the stage that fits — or read the whole ladder below.
The Kiselev course is complete: all twelve of his school texts are now in English.
How old is the student?
What have they already worked through?
Ages are the usual range, not a rule. Adults working alone start wherever the contents look unfamiliar — the tables of contents on each book page are there for exactly that.
The series at a glance
Every stage, the text or texts that carry it, and the problem collections that go alongside. Where a stage lists more than one text, they cover the same ground different ways — take one.
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Kiselev's Systematic Arithmetic · Kiselev's Arithmetic · Malinin & Burenin · Rashevsky's Arithmetic — take one
with Malinin & Burenin Problems · Malinin's Mental Calculation · Vereshchagin's Arithmetic Problems · Berezanskaya's Arithmetic Problems · Rachinsky Solutions · Rachinsky's 1001 Problems · Perelman's Entertaining Arithmetic · Berman's Calculation Techniques
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Elementary Algebra · Malinin and Burenin's Algebra · Rashevsky's Elementary Algebra · Davidov's Algebra · Barsukov's Algebra · Kiselev's Algebra I — take one
with Lebedintsev's Algebra Problems, Part I · Shaposhnikov & Valtsov, Part I · Kiselev's Algebra Problems
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Elementary Geometry · Davidov's Geometry · Kiselev's Planimetry · Rashevsky's Geometry · Perelman's Geometry — take one
with Delone: Geometry Problems · Rybkin's Planimetry Problems · Aleksandrov's Geometric Problems · Geometry Through Problems
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Elements of Algebra & Analysis I · Kiselev's Algebra II — take one
with Shaposhnikov & Valtsov, Part II · Kiselev's Algebra Problems
- 5. Trigonometry 15+
Derzhavin's Trigonometry · Anoshchenko's Trigonometry · Davidov's Trigonometry · Rybkin's Plane Trigonometry · Rashevsky's Trigonometry — take one
with Vereshchagin's Plane Trigonometry · Rybkin's Trigonometry Problems · Schiff's Trigonometry Problems
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with Delone: Geometry Problems · Rybkin's Stereometry Problems · Rybkin's Trigonometry Problems
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Elements of Algebra & Analysis II · Kiselev's Calculus — take one — Elements II for a reader who has worked through Elements I
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Gunter & Kuzmin · Gunter & Kuzmin, Part I · Gunter & Kuzmin, Part II — the combined volume, or its two halves in order
with Demidovich's Analysis Problems · Demidovich's Analysis Problems for Engineers · Markov: Probabilities
The whole sequence
Arithmetic
Whole numbers, divisibility, fractions, decimals, ratio and proportion — every rule derived rather than asserted.
The text
Kiselev's Systematic Arithmetic
by A. P. Kiselev
US grades 5–8 · UK Years 6–9
Kiselev's definitive systematic arithmetic — the expanded companion to his classic arithmetic textbook. 207 pages, 264 numbered articles across seven divisions: whole numbers; concrete (denominate) numbers and the metric system; divisibility, primes, and factorisation; common fractions; decimal fractions with limits and repeating decimals; ratio and proportion; and a full course of word problems on proportional quantities. Includes elementary treatments most textbooks leave out — Euclid's proof of the infinitude of primes and the concept of a limit used to prove that 0.999… equals 1.
Kiselev's Arithmetic
by A. P. Kiselev
US grades 4–7 · UK Years 5–8
The foundational arithmetic textbook used in Russian schools for over a century. 6 parts, 214 sections. Whole numbers through proportional quantities, with mathematical proofs for every operation.
Malinin & Burenin
by A. Malinin & K. Burenin
US grades 5–8 · UK Years 6–9
A complete English translation of Malinin and Burenin's Imperial Russian arithmetic textbook. 11 chapters, 301 sections. Covers numeration, operations on whole numbers, divisibility, fractions, ratios, proportions, percentage, interest, and Diophantine problems.
Rashevsky's Arithmetic
by K. N. Rashevsky
US grades 5–8 · UK Years 6–9
A complete course in arithmetic in nine sections, in first faithful English translation from the original Russian edition. The four operations on whole numbers with the written methods set out in full; named and metric quantities; areas and volumes worked from drawings; tests for divisibility, primes, greatest common divisor and least common multiple; common and decimal fractions, including repeating decimals; ratio, proportion and proportional division; percentage and mixing problems; and plans, diagrams and graphs. Long divisions and column multiplications are printed digit under digit, so the reasoning stays visible on the page.
Problems and enrichment alongside it
Algebra, first course
Algebraic notation, signed numbers, equations and inequalities, polynomials, fractions, ratio and proportion, coordinates and graphs.
The text
Elementary Algebra
by A. P. Kiselev
US grades 7–11 · UK Years 8–12
The single-volume algebra course, covering in one book essentially what Algebra Part I and Part II cover between them. Concepts arrive in a deliberate order, rules are tied to the operations that justify them, and proof is treated as part of elementary education rather than an ornament. Keeps hand extraction of cube roots, indeterminate multipliers and compound interest; omits the functions-and-graphs material of the two-part course.
Malinin and Burenin's Algebra
by A. F. Malinin & K. P. Burenin
US grades 8–12 · UK Years 9–13
The algebra course and its problem collection in one volume, in first faithful English translation from the original Russian edition. An Introduction and sixteen chapters, 429 numbered articles, run from positive and negative numbers and the operations on algebraic expressions through equations of the first degree in one and several unknowns; powers and roots, with the square and cube roots of numbers and the square root of polynomials; quadratic equations, imaginary expressions and the trinomial of the second degree; inequalities; indeterminate equations; proportions; arithmetic and geometric progressions; logarithms; combinations and Newton’s binomial; continued fractions; the method of limits; and series. Problem sets follow the articles throughout — more than 3,300 numbered problems, 620 of them in a closing chapter of miscellaneous problems — and 2,698 answers are printed at the back.
Rashevsky's Elementary Algebra
by K. N. Rashevsky
US grades 8–11 · UK Years 9–12
A complete course in school algebra, taught with the graph as a working instrument, in first faithful English translation from the original Russian edition. An Introduction and nineteen chapters, 308 numbered articles, run from the use of letters and signed numbers through operations on expressions, factoring, algebraic fractions, negative exponents and proportions; equations of the first degree in one and several unknowns and their investigation; functions of the first degree and their graphs; powers and roots, square roots of numbers, irrational numbers and expressions, and the generalized concept of a power; the quadratic equation, functions of the second degree and the graphical solution of quadratics; imaginary numbers and their representation on the plane; irrational and higher-degree equations and systems of the second degree; progressions; logarithms with the use of tables, exponential equations and compound interest; and groupings, Newton’s binomial and a short historical sketch of algebra. Forty-nine figures carry the graphical method from the first chapter on. Six short exercise sets are included, without answers.
Davidov's Algebra
by A. Yu. Davidov
US grades 8–12 · UK Years 9–13
The algebra course written by the founding president of the Moscow Mathematical Society, in first faithful English translation from the original Russian edition. Five divisions and 478 numbered articles run from algebraic signs and the fundamental operations through polynomials, the greatest common divisor, algebraic and continued fractions, ratio and proportion; equations of the first degree in one, two and several unknowns; powers and roots, irrational and incommensurable quantities, Newton’s binomial, quadratic equations and their investigation, imaginary quantities and equations of the third and fourth degree; indeterminate analysis; and progressions, series, logarithms and compound interest. Full solutions to the problems are included.
Barsukov's Algebra
by A. N. Barsukov
US grades 8–10 · UK Years 9–11
The Soviet school's standard algebra textbook — ten editions, roughly ten million copies. The first complete, faithful English translation of Barsukov's course, prepared from the original Russian and typeset to modern standards with source-exact section numbering. Twelve chapters, 130 numbered sections: from algebraic expressions and rational numbers through factoring, algebraic fractions, coordinates and graphs, systems of equations, the square root worked digit-by-digit, quadratic equations, and functions and graphs. Includes a full chapter on the slide rule — a subject once central to technical education, almost never found in a modern Western algebra text.
Kiselev's Algebra I
by A. P. Kiselev
US grades 7–9 · UK Years 8–10
6 chapters, 126 sections, 239 exercises. From relative numbers and polynomials through first-degree equations, powers, roots, and quadratics.
Problems and enrichment alongside it
Geometry of the plane
The straight line, the circle, similar figures, regular polygons, areas — proof-based from the first page.
The text
Elementary Geometry
by A. P. Kiselev
US grades 8–11 · UK Years 8–13
The complete geometry course — five books of plane geometry and three of solid geometry — in one English volume. Definitions are precise, theorems are proved, constructions are justified, and each idea is prepared by the one before it. 380 pages and 450 figures.
Davidov's Geometry
by A. Yu. Davidov
US grades 8–11 · UK Years 9–12
The geometry course written by the founding president of the Moscow Mathematical Society, in first faithful English translation from the original Russian edition. Two parts, thirteen chapters and 305 numbered sections. The planimetry treats straight lines and angles; figures, the equality of triangles, perpendicular and oblique lines; parallel lines, parallelograms and trapezoids; similarity and the ratio of lines; the circle — chords and tangents, the measurement of angles, proportional lines, inscribed and circumscribed polygons, the relative position of two circles, the four remarkable points of a triangle, reciprocal points and polars; regular polygons; the measurement of areas, with the surveying of a plan; and the circumference and area of the circle by limits. The stereometry covers lines and planes in space; dihedral, polyhedral and trihedral angles; prisms, pyramids, symmetric and regular polyhedra; volumes; the cylinder, cone and sphere, the spherical triangle, and the conic sections. 344 figures and 256 problems, with solutions to all of them. A digital edition, read online in Gumroad Reader.
Kiselev's Planimetry
by A. P. Kiselev
US grades 8–10 · UK Years 8–11
The first complete English translation of Kiselev's proof-based Euclidean geometry. 269 sections across 6 chapters: basic concepts, triangles, circles, similarity, regular polygons, and areas. PDF, paperback, and hardcover.
Rashevsky's Geometry
by K. N. Rashevsky
US grades 8–11 · UK Years 9–12
A complete school course in plane and solid geometry in one volume, in first faithful English translation from the original Russian edition. Two parts, thirty chapters and 406 numbered articles. The planimetry runs from the straight line, polygons and the circle through the three cases of equality of triangles, parallel lines, parallelograms and trapezoids and construction problems; the method of loci, parallel translation and rotation; ratio and common measure, angles in a circle, proportional segments, similarity and homothety; the basic trigonometric functions of the right triangle; regular polygons, areas, the concept of a limit, the circumference and area of the circle, and algebra applied to geometry. The stereometry covers the position of a plane, perpendicular and parallel lines and planes, dihedral and polyhedral angles, polyhedra and their volumes, the cylinder, cone and sphere, incommensurable quantities, and closes with practical work: making a plan and levelling. 338 figures and 367 exercises in ten graded sets.
Perelman's Geometry
by Ya. I. Perelman
US grades 7–10 · UK Years 8–11
A practical course in plane and solid geometry for self-study, carried as far as the trigonometry of the right triangle, in first faithful English translation from the original Russian edition. Nine chapters: the straight line and the circle, with scales, circumferential speed, belts and gear wheels; angles and parallel lines, with the protractor; the triangle, its construction and the tests for equality; polygons and the parallelogram, with the vernier and slide calipers; chords, tangents, conjugation and the ellipse; similarity, Pythagoras’ rule and the square root; tangent, cotangent, sine and cosine, with their tables; areas, from the rectangle to the circle, the ellipse and irregular figures; and the surfaces, volumes and weights of solids. The main text rests on about forty theorems; 458 problems, about half the book, carry the rest, with solutions or answers to 423 of them. 230 figures.
Problems and enrichment alongside it
Algebra, second course
Powers and roots, quadratics, progressions, logarithms, combinations and the binomial theorem, limits.
The text
Elements of Algebra & Analysis I
by A. P. Kiselev
US grades 8–11 · UK Years 9–12
Part One builds a connected foundation: algebraic notation and operations, signed numbers, identities, equations and inequalities, polynomials, fractions, ratio and proportion, functions, coordinates and graphs, powers, roots and irrational numbers, progressions, logarithms with four-figure tables, compound interest and the binomial theorem. A distinct work from the 1888 Elementary Algebra.
Kiselev's Algebra II
by A. P. Kiselev
US grades 9–12 · UK Years 10–13
12 chapters plus Supplements, 187 sections. Functions, quadratic systems, inequalities, logarithms, progressions, complex numbers, the binomial theorem, and continued fractions.
Problems and enrichment alongside it
Trigonometry
Goniometry, the trigonometric functions and their identities, triangle solution, trigonometric equations.
The text
Derzhavin's Trigonometry
by S. S. Derzhavin
US grades 10–12 · UK Years 10–13
A complete classical course in plane trigonometry, in first complete English translation. Eight divisions and 86 numbered sections run from the function concept and radian measure through the trigonometric functions traced quadrant by quadrant, the addition and reduction formulas, the construction and use of logarithmic tables, the solution of oblique triangles by the laws of sines and cosines and by Mollweide’s formulas, measurement in the field and triangulation, the computation of π, and trigonometric equations. Sixty-eight figures and Tables I–V.
Anoshchenko's Trigonometry
by P. M. Anoshchenko
US grades 10–12 · UK Years 10–13
The 1905 secondary-school course in plane trigonometry, in first faithful English translation from the original Russian edition. Three parts: goniometry builds the measurement of angles and the trigonometric quantities from first principles across twelve chapters; rectilinear trigonometry proper solves the right triangle and then the oblique triangle, and closes with a full chapter on field triangulation; supplementary articles take up trigonometric equations, inverse trigonometric quantities, and maxima and minima. Sixty-three figures on eight plates.
Davidov's Trigonometry
by A. Yu. Davidov
US grades 10–12 + · UK Years 10–13
Plane and spherical trigonometry in one course, by the founding president of the Moscow Mathematical Society, in first complete English translation. Three parts and 97 numbered sections. Goniometry builds the trigonometric lines from geometric construction, follows them through the whole circle, and treats the addition, multiplication and division of arcs and the construction and use of trigonometric tables, Lalande’s and the seven-digit tables among them. Plane trigonometry solves right and oblique triangles and applies them to measurement: inaccessible distances and heights. Spherical trigonometry derives the fundamental formulas, Delambre’s formulas and Napier’s analogies, solves right and oblique spherical triangles and finds the area of a spherical triangle. A seven-part Supplement takes up inverse trigonometric functions, De Moivre’s formula, the roots of xᵐ = 1, regular polygons, the cubic equation solved by trigonometric lines, and the expansion of the functions into series. Sixty figures, and 265 problems, 212 of them answered in the key at the back.
Rybkin's Plane Trigonometry
by N. A. Rybkin
US grades 10–12 · UK Years 10–13
A complete course in plane trigonometry, in first faithful English translation from the original Russian edition. Eleven chapters and 126 numbered sections in three divisions. Goniometry builds the six trigonometric functions from lines in the circle, first for the acute angle and then through the full revolution: reduction formulas, negative angles and periodicity, graphs of the functions, inverse circular functions, the sum and difference, double-angle and half-angle formulas, reduction to logarithmic form and the auxiliary angle, and a chapter of trigonometric equations that shows how roots are lost. The second division explains how trigonometric tables are constructed and how four- and five-place tables are used. The third solves right and oblique triangles, each case worked with natural values and again with logarithms, and closes with measurement in the field: inaccessible distances, heights and triangulation. Fifty-four figures and a summary of the basic formulas.
Rashevsky's Trigonometry
by K. N. Rashevsky
US grades 10–12 · UK Years 10–13
A faithful English edition of the classical Russian trigonometry textbook. A complete course in plane trigonometry — the trigonometric functions, radian measure and the unit circle, identities, equations, and the solution of triangles — built the way the Russian school tradition built it: define, prove, apply.
Problems and enrichment alongside it
Geometry of space
Lines and planes, dihedral and polyhedral angles, the regular polyhedra, prisms, pyramids, cylinders, cones, the sphere, volumes.
The text
Problems and enrichment alongside it
Introduction to analysis
Limits, the derivative, maxima and minima, the beginnings of the integral calculus.
The text
Elements of Algebra & Analysis II
by A. P. Kiselev
US grades 11–12 · UK Years 12–13
Part Two carries the numbering of Part One forward into analysis: the theory of limits and its applications to elementary geometry, derivative functions, slopes of straight lines and curves, increasing and decreasing functions, and maxima and minima.
Kiselev's Calculus
by A. P. Kiselev
US grades 11–12 · UK Years 12–13
The first complete English translation of Kiselev's calculus — the final volume in his school course. Single-variable differential and integral calculus, proof-based, with classical applications to mechanics and physics. The bridge from Algebra II into the analysis tradition that produced Markov's probability theory.
University mathematics
Analytic geometry, differential and integral calculus, differential equations, series, complex variables, probability.
The text
Gunter & Kuzmin
by N. M. Gunter & R. O. Kuzmin
US grades University 1–4 · UK Undergraduate
The standard Russian problem collection for higher mathematics — the book used across Soviet technical universities. 814 pages spanning mathematical analysis, differential and integral calculus, series, differential equations, and probability theory, in the first complete English translation.
Gunter & Kuzmin, Part I
by N. M. Gunter & R. O. Kuzmin
US grades University 1–2 · UK Undergraduate
The first half of the standard Russian problem book for higher mathematics, as a volume on its own: analytic geometry in the plane and in space, differential calculus and its applications including differential geometry, higher algebra, and indefinite and definite integration. 448 pages.
Gunter & Kuzmin, Part II
by N. M. Gunter & R. O. Kuzmin
US grades University 2–4 · UK Undergraduate 2–3
The second half completes the collection: multiple, curvilinear and surface integrals, field and potential theory, ordinary and partial differential equations, improper integrals and special functions, series and Fourier series, approximate computation, functions of a complex variable, mathematical physics, the calculus of variations and probability. 379 pages.
Problems and enrichment alongside it