Kiselev's Calculus is now in English

The calculus textbook from the tradition that trained the Soviet mathematical school — Kolmogorov, Arnold, Gelfand, Manin, Sinai — completes the Kiselev English Editions series. After 117 years, the original 1909 calculus is now available in a complete English translation.

Friends,

A short post today, to share that the fourth volume is done.

Kiselev's Elements of Differential and Integral Calculus — first published in 1909 — is now available as a complete English translation. It is the fourth and (for the algebra/calculus arc) final volume in the Kiselev English Editions series, after Arithmetic, Algebra Part I, and Algebra Part II.

For most of the twentieth century, the Russian mathematical tradition produced more research-active mathematicians than any other national school. Kolmogorov, Arnold, Gelfand, Manin, Sinai — and three Soviet Fields Medalists — all came up through the same school system. The textbooks of that system were Kiselev's: arithmetic, algebra, geometry, and calculus. Every key Soviet mathematician, physicist, and engineer of the twentieth century met the elements of calculus the same way — through this book or its lineal descendants.

That book is now in English.

What is in the calculus volume

Seven chapters. 110 numbered sections. 41 figures. 214 exercises with answers.

The chapters, in order: foundations of the theory of limits; some applications of the theory of limits; the definition of the derivative and the differential, together with the rules of differentiation and the derivatives of the elementary transcendental functions; the geometric and mechanical meaning of the derivative; applications of the derivative to the investigation of functions (maxima and minima, curve tracing, Rolle's theorem, Lagrange's Mean Value Theorem); the definite integral developed from the area problem and the path problem; and the elementary methods of integration.

This is a textbook for a student who has finished secondary algebra and is meeting calculus for the first time. It is also, in my view, the cleanest first encounter with the subject available — including in comparison with modern textbooks at three times the length.

Two examples make this concrete.

The limit, before the derivative

Most modern American calculus textbooks introduce the derivative in Chapter 2, then in Chapter 3 introduce limits as the machinery that makes the derivative work. The structure is reversed from how the subject was actually built historically, and — in practice — most students never recover from the inversion. They learn to compute derivatives mechanically, never having internalised what a limit is.

Kiselev's Chapter 2 — fifteen pages — develops the theory of limits first. It defines an infinitesimal as a variable quantity whose absolute value becomes and remains smaller than any positive number, however small. It proves that the algebraic sum of a finite number of infinitesimals is an infinitesimal. It proves that the product of an infinitesimal by a finite quantity is an infinitesimal. It defines the limit of a variable: a is the limit of x if the difference a − x is an infinitesimal. From this it proves the limit of a sum, product, and quotient. It proves that a monotone bounded sequence has a limit. It proves the sandwich theorem.

Only after all of this — eighteen sections in — does Kiselev open Chapter 4 with the definition of the derivative as the limit of the difference quotient. The student arrives at the derivative with a working notion of limit already in hand. This is the order of historical development, and it is also the order in which the human mind absorbs the subject most easily.

When the limit is properly internalised first, the derivative becomes obvious. When it is not, the derivative becomes mechanical.

The proof that lim(sin α)/α = 1

The second example: the most cited "trust me" limit in every American calculus course.

Modern American textbooks present this limit and assert it without proof. The student is expected to take it on faith and apply it. Some textbooks gesture at "it can be shown geometrically." A few include a sketch in the back matter.

Kiselev's Chapter 3, §62 — labeled "Lemma" — opens with the statement and immediately gives the proof. Five lines. The construction: a circular arc of radius 1, the angle α at the centre, the inscribed chord (length 2 sin(α/2), so for the half-angle, length sin α), and the circumscribed tangent (length tg α). The arc is between the chord and the two tangent segments. Therefore sin α < α < tg α. Dividing through by sin α and taking the limit as α tends to zero, the squeeze theorem (just proved in Chapter 2) gives lim(sin α)/α = 1.

The proof is accessible to a 17-year-old. It is the proof that should be in every calculus textbook. It is not in most of them.

What this tells you about Kiselev: he doesn't skip the parts that are inconvenient. He does the work, even when the work is short and unglamorous. The student is not asked to take anything on faith.

The English edition

For most of the last century, Kiselev's calculus has been inaccessible to English-speaking students. The algebra volumes — Parts I and II — were published in English earlier this year. Calculus is the fourth volume; with it, the algebra/calculus arc of the Kiselev English Editions series is complete.

The translation preserves Kiselev's original section numbering, the order of his exposition, and the precision of his prose. The typography is modern; the mathematics is Kiselev's. Notation is presented as Kiselev presented it: Log for the natural logarithm, tg and cotg for the tangent and cotangent, the historical 2d for two right angles in the relevant passage — each with a translator footnote on first use.

Calculus is now on Gumroad as a PDF and on Amazon as a paperback. Hardcover and Kindle editions are on their way to Amazon — uploaded for KDP review, expected live in the next few days.

If your child, your student, or you would like to learn — or relearn — calculus from the textbook by which generations of Russian and Soviet schoolchildren met the subject, there is now no language barrier.

Get Calculus on Amazon (paperback, live now): a.co/d/0dSQmJRr

Get Calculus on Gumroad (PDF): valeman.gumroad.com/l/k…

The earlier Kiselev volumes — Arithmetic, Algebra Part I, and Algebra Part II — along with other translated Russian mathematics textbooks, are at russianmathbooks.com.

The Geometry — Planimetry and Stereometry — comes next.

— Valeriy