You are here


Israel M. Gelfand and Tatiana Alekseyevskaya
Publication Date: 
Number of Pages: 
[Reviewed by
Bill Satzer
, on
This introduction to plane geometry is the fifth book written by Israel Gelfand and a variety of co-authors aimed initially at high school students. (The others are The Method of Coordinates, Functions and Graphs, Algebra, and Trigonometry.) Each of these was intended to teach mathematical notions in a fresh way to students and general readers who might not have any familiarity with the subject.
Unlike many introductions to geometry that focus on logic, axioms, and proving theorems, this one takes quite a different path. Geometry here is more about geometrical constructions and concepts that are developed visually. From the introduction: “the main goal is to understand the structure of our space, learn how to see it, and find out how to orient ourselves in it.” This book is intended to engage the reader visually, tactilely, and kinesthetically. Theorems and proofs appear throughout, but they are mainly used to underline and amplify what is discovered visually.
The dominant features of this book are its figures – all geometric constructions, and more than 400 of them. In many respects they are absolutely the heart of the book. Yet there are so many that they are almost overwhelming. They are very simple figures to start with – just points and lines – but they become more and more complicated.
The book introduces geometric concepts very slowly in the beginning. It starts with the simplest of geometric features and adds elements very parsimoniously. Only in the last part of the book does the student see anything numerical such as length, area, or angular measure. The first part is about points and lines; the authors call it “a look at projective geometry”. The authors move on to affine geometry in the second part by adding the notions of parallel lines and parallel transport. Then “symplectic geometry” (as an optional topic) comes in as the “area of a figure” without an area measure. The idea here is describe what properties area should have and then to assess area by comparing figures with a unit parallelogram. Finally, length and angular measure are introduced in the final and longest part with most of the other elements of ordinary Euclidean geometry.
Although this is intended as an elementary introduction, it teaches some sophisticated subjects and does that in novel ways. Already early in the first part the authors introduce Desargues configurations of triangles and the idea of a convex hull, for example. The last chapter has a much more extensive treatment of circle geometry than is common in elementary books, and a good deal about inscribed and circumscribed polygons.
This book was completed before Israel Gelfand died, but it took several years for the coauthor to prepare all the drawings so they could be reproduced accurately in print as they were originally drawn. Tatiana Alekseyevskaya was the primary writer; she says that Israel Gelfand thought of himself as a conductor or composer who set out the important themes and established the framework. But it was up to her to do all the hard work. (She never says this, but it’s quite evident.)
Although this could be used as a textbook as the authors suggest, its best use might be as supplementary material for an instructor presenting a course for teachers. It has a good set of material to enliven more traditional geometry instruction.
The book has a complete glossary but no index. There are problems and exercises throughout. The exercises are accompanied by solutions.


Bill Satzer (, now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications ranging from speech recognition and network modeling to optical films, material science and the odd bit of high performance computing. He did his PhD work in dynamical systems and celestial mechanics.