The book under review is an expanded translation of a unique phenomenon in the Russian mathematical literature. First published in 1892 by A. P. Kiselev as *Elementary Geometry*, by 1940 it underwent more than 40 revisions and eventually became a measuring rod for geometry education in Russia against which all other textbooks had to be judged. If nothing else, this book’s staying power may serve as an enticement to anyone interested in, or involved with, high school geometry. Its introduction to the English speaking student and teacher is thus more than welcome. The effort by A. Givental, who translated the book from Russian and combined pieces of the many editions of the original, deserves wholehearted recognition and sincere praise.

The early history of the book is murky. In the Tsarist Russia *Kiselev’s Geometry* competed successfully against other textbooks. Its 23rd edition (1914) is available online. The upheaval of 1917 brought an overhaul of the education system based more on revolutionary zeal than on evolutionary societal demands. But towards the early 1930s the situation was ripe for a more rational attitude. On February 12, 1933, the Central Party Committee has issued a directive that instructed the responsible organizations to replace the "working (class) books" used in schools until then with specially designated "stable" textbooks (Math Education, n1 1934, p. 71). On November 16, 1936, the Mathematical Committee of the People’s Commissariat of Education has branded the originally selected geometry book as an illiterate concoction of the lowest quality and recommended its immediate removal. A meeting of the Moscow Mathematical Society held on April 9, 1937 accepted a resolution expressing outrage that the book had been yet republished for another year and recommended *Kiselev’s Geometry* as a temporary replacement. The new 41st edition of *Kiselev’s Geometry* was edited by N. A. Glagolev.

The "temporary replacement" kept the official title until the mid 1950s. It appears that in 1955 the recognition as official text was awarded to the book by N. N. Nikitin and A. I. Fetisov. However, already in 1956 regrets were sounded, and for the first time in 20 years it was noted that the Department of Education and the UchPedGiz (the official textbook publishing agency) had all along been giving a wrong interpretation of the Central Committee’s directive, as if it really stipulated *a* unique stable textbook for every subject. *Kiselev’s Geometry* was allowed to stay around together with the new text until the latter underwent serious reworking.

In time, *Kiselev’s Geometry* stood its ground against textbooks by N. A. Izvolsky, V. G. Boltyansky and I. M. Yaglom, A. N. Kolmogorov, a translation of one by J. Hadamard, a book written jointly by Nikitin and Fetisov, and then the books they wrote separately, and many others. The friends I polled who took geometry in the early 1960s all used Kiselev’s book ,with the exception of experimental classes in special math schools that used the book by Fetisov. In later years, there was serious competition from Nikitin’s book but *Kiselev’s Geometry* kept going on even if as a backup text. I do not know whether at that time any book had been assigned official status, but in a booklet on *Proofs in Geometry*, first published in 1977, A. I. Fetisov, while illustrating an erroneous proof notes dejectedly that the diagram is similar to the one used in an officially "approved book". There is no doubt to about which book he meant.

From the discussions of 1955-1956 it appears that *Kiselev’s Geometry* was judged to follow Euclid’s *Elements* too closely. As one example, it was pointed out that, by introducing the parallel postulate early on, the book by Nikitin and Fetisov managed to simplify the treatment of the Exterior Angle Theorem compared to its treatment in *Kiselev’s Geometry*. There is no doubt that the wind of change in geometry education that began blowing in Europe at the end of the 19th century, had reached and influenced the Russian policy makers. This particular instance is usually cited as a great insight of Euclid’s who postponed using the Fifth Postulate until I.29. He accordingly split the Exterior Angle Theorem into two parts, the first of which did not require parallel lines. The true appreciation of Euclid’s sequence came with the 19th century development of various geometries, and "Absolute Geometry" in particular. To a teacher, a theorem of such significance provides a rich background for historical and philosophical discussion that most students are capable of appreciating.

In this context, Kiselev follows in Euclid’s footsteps and delays introducing the Parallel Postulate until the second quarter of the book. (Unfortunately, he also repeats Euclid’s error while proving the absolute part of the theorem.) He does not follow Euclid blindly, though. Geometric objects are introduced each at its time, not at the beginning of the book. Their properties are defined when they become needed or are about to be proved. Kiselev also goes to some length to clarify general notions, like theorem and axiom, explains the relation between a theorem and its converse, inverse, and contrapositive statements, and a proof by contradiction. Many sections are preceded by short introductions.

Sometimes the divergence from the *Elements* is significant; for example, in his treatment of the Pythagorean theorem. He proves the theorem only after the theory of proportions. And the proof (p. 152) is a simplified version of Euclid’s VI.31, to which the translator gives special attention towards the end of the book. Along with the full version of VI.31 we find there Euclid’s I.47 and a proof by rearrangement. Kiselev’s is, in my opinion, the best selection among a great variety of the proofs of the famous theorem.

The book was originally written in a clear, no-nonsense style which has been polished over its many editions and revisions. The style was well preserved in the translation. There is nothing in the book that will even occasionally distract from the subject.

The most shining example is Kiselev’s treatment of the circumference and π. Circumference is defined (p.202) as the limit of the perimeters of regular polygons inscribed into the circle when the number of sides is doubled indefinitely. Doesn’t that sound impressive? But consider then that Kiselev goes precedes the definition with a reasonably developed theory of limits (pp. 195-199). The book gives a very clear introduction into the theory of limits which, although not 100% rigorous, neither noticeably cuts corners. For example, the book does prove that the limit (which is the circumference) exists (p. 202). This comes after a thorough discussion of similarity (pp. 117-182). In particular, he proves (p. 187) that regular polygons with the same number of sides are similar, and their sides have the same ratio as their radii or apothems. Combining this with the theory of limits he derives (p. 204) the fact that the ratio of the circumference to the diameter is the same for all circles and thus introduces \(\pi\).

Simple as this may appear, even the best of the modern high school geometry texts make a mess of the definition of \(\pi\). For example, H. Jacobs (*Geometry*, 3rd ed, p. 592) defines it as the limit, as the number of sides increases, of the ratio of the side to twice the radius of a regular polygon. He does this after constructing a table for several polygons and observing the behavior of the ratio. Thus the definition of \(\pi\) and the determination of its value are combined into one, while the reason for the existence of \(\pi\) is not discussed at all.

Another topic that drew my attention is the irrationality of the square root of 2. The proof given here is remarkably similar to the one recently published by Tom Apostol (*The American Mathematical Monthly*, v 107, n 9, pp 841-842). Unlike Apostol’s, Kiselev’s proof does not qualify as a proof without words, but it appeals to the same geometric ideas. However, this is not exactly the reason I like Kiselev’s treatment. Kiselev precedes his proof with the notion of mensuration and a complete discussion of the Euclidean algorithm including Archimedes’ Axiom. The proof is then followed by articles on lengths of segments, approximation, irrational numbers and the number line.

In the Translator’s Foreword, Professor Givental mentions three virtues of a good textbook (precision, simplicity, conciseness) formulated yet by Kiselev himself, and adds a fourth one — competence in the subject. Thinking specifically of *Kiselev’s Geometry*, one other feature must be mentioned: *autonomy of the discourse*.

Every textbook is created for a particular audience which is usually characterized by the level of preparedness to absorb the material, both in terms of the requisite knowledge and the ability to do so. The requirements are usually set up in the introduction and are commonly violated in the text. This is either done tacitly or with a reference to the imposed limitations on the size or the scope of the book. One salient virtue of *Kiselev’s Geometry* is that, throughout, the author remains faithful to his intended audience, viz., middle and junior high school students taking up geometry for the first time. Assuming only very basic knowledge of mathematics, Kiselev builds the edifice of geometry from the bottom up, supplying both bricks and mortar in the process. The book is very much self-contained. The only outside references come at the upper storey, as for example, when he only mentions Gauss’ constructibility theorem for regular polygons.

The book comes with nearly 600 exercises, which are distributed throughout. Some are supplied with hints but none with a solution. I do not believe Kiselev had any intention or a pedagogical reason to conceal solutions from the student. In the Introduction to the first edition, he mentions a then available problem collection from which he drew exercises. In my time, another problem collection, with all the exercises solved, was used alongside *Kiselev’s Geometry*. While I do think that the absence of a solution key in the first English edition may deter some potential users, I do not believe it should. On one hand, in the body of the book, Kiselev devotes considerable time to solving problems, paying special attention to a variety of basic constructions. The text is interspersed with remarks on problem solving and methods of proof all of which come with practical demonstrations. On the other hand, at this time of easy access to the internet, an interested student can get a solution to a problem or two or find some advice to help with a solution. There are a multitude of online forums that exist just to this end.

The book is light and relatively small, but due to Kiselev’s concise, no-frills style, the topics covered are awarded comprehensive treatment. The book will give good service to geometry students and teachers, homeschoolers, student teachers and their instructors. All should be looking forward to the appearance of the stereometry part of *Kiselev’s Geometry*.

Alex Bogomolny is a business and educational software developer who lives with his wife and little son in East Brunswick, NJ. In August 2006 his popular web site Interactive Mathematics Miscellany and Puzzles has welcomed its 20,000,000th visitor.