Isaac M. Yaglom, an influential figure in mathematics and mathematics education in the Soviet Union during the 20^{th} century, published his two volume *Geometric Transformations* in Russian in 1955 and 1956. The first volume was translated into English by Allen Shields and published as two separate texts in 1962 and 1968. The first half of the second volume was translated into English by Abe Shenitzer in 1973. These first three books (*Geometric Transformations I, II, **III*) are currently available as Volumes 8, 21, and 24 of the MAA’s *Anneli Lax New Mathematical Library*.

We had to wait over 35 years for the last portion of Yaglom’s impressive work to be available in an English translation: *Geometric Transformations IV* was published in 2009 as Volume 44 of the *New Mathematical Library*. Abe Shenitzer again served as the translator, producing a text that in language bears no hint of a non-English origin (the previous three volumes are also excellent translations). We owe a debt of gratitude to Shenitzer and the MAA for finally making available the complete *Geometric Transformations* to English readers — it is a unique and beautiful four volume series. (All references in Volume III to “untranslated Russian material” can now be interpreted to refer to Volume IV.)

The books fit perfectly into the *New Mathematical Library*, a series of monographs intended not as textbooks but rather as well-written supplements for high school or early college students on a variety of topics not usually seen in the standard curriculum. A goal for the NML monographs is to require of readers little prior technical knowledge but much disciplined intellectual effort.

Yaglom’s *Geometric Transformations* was written five years prior to the establishment of the NML series, but the goal for this work fits perfectly with the NML philosophy. Assuming only a background of high school Euclidean plane geometry, Yaglom develops the ideas inherent in the *group-theoretic foundation of geometry* as formulated via *geometric transformations*. His primary audience was intended to be high school and early college students. He uses a purely synthetic approach — no vectors or linear algebra will be found in these books — with a rich collection of illustrations. (One annoying weakness: none of the volumes have indexes and the tables of contents are brief.)

As divided up for the NML series, the four volumes consider the following topics in plane geometry:

I: Isometries (distance preserving transformations).

II: Similarities (shape preserving transformations).

III: Affine and projective transformations, and hyperbolic geometry (Klein disk).

IV: Conformal and circular transformations, and hyperbolic geometry (Poincaré disk).

Each book has a clear exposition of the core material, with between 50 to 100 exercises embedded in the text. These are not routine, computational problems — they require the reader to tackle interesting but difficult geometric questions. The nature of the exercises changes over the course of the four NML volumes: most from the first two volumes ask the reader to carry out various geometric constructions, while those from the last two usually ask for proofs of theorems. This reflects an increasing level of sophistication in the material between the beginning and end of the series.

The exercises illustrate the purpose and power of the ideas developed in the exposition via *application* of the ideas. Cultivating such understanding and skill is not given enough emphasis in some geometry texts: they establish powerful theoretical results but use them to solve only a minimal number of geometric problems. As one reader on Amazon.com observed, Yaglom’s books “…are not primarily about transformations, they are mainly about how to use transformations to solve problems.”

To further support this function, almost half of each volume is devoted to detailed solutions of all the embedded exercises. It’s clear that the *Geometric Transformations* series is an invaluable resource for honing geometric problem solving skills, be that for serious mathematical competitions such as the Mathematical Olympiad or the Putnam exam, or just for personal challenge and enjoyment.

As Yaglom makes clear in the introduction to Volume III, though he deals with various groups of transformations (e.g., isometries, projective transformations), he is not trying to present an introduction to the geometries associated with the groups. Instead, his aim is to “…demonstrate that the existence of various geometries can be very helpful even if we do not go beyond elementary geometry. If we realize that a particular theorem is essentially a theorem of, say, projective geometry (that is, the theorem deals with properties unchanged under projective transformations), then we can frequently simplify its proof.” The one exception is Yaglom’s treatment of hyperbolic geometry: he describes this geometry in two Supplements to Volumes III and IV, the first discussing the Klein disk model and the second the Poincaré disk model.

Each of the first three volumes starts with an Introduction titled “What is Geometry?” These three essays build on each other and on the material Yaglom has already developed, and constitute a lovely elementary description of the basic tenet of Felix Klein’s Erlanger Programm. The first essay, starting from very intuitive ideas, winds its way to Yaglom’s first definition of geometry: “*Geometry is the science that studies those properties of geometric figures that are not changed by motions of the figures.*” Here a *motion* is a geometric transformation preserving distance between points, i.e., an *isometry*. By the third essay Yaglom has developed enough material to state his more general definition of geometry: “*A geometry is a discipline concerned with those properties of figures which do not change under the transformations of a group of transformations.*” By this time the readers have been introduced to the group of isometries and the group of similarities, and understand that each defines a different geometry on the plane. Yaglom can now expand further: to affine and projective transformations, to hyperbolic transformations, and to circular transformations.

The recently released Volume IV is broken into five chapters, a supplement, and a collection of solutions to all eighty-nine problems embedded in the text. The first four chapters are concerned with *inversions*, also termed *reflections in circles* in the plane. Chapter 1 gives the definition of inversion (well motivated and compared with reflections in lines) and establishes the basic properties, primarily that an inversion (A) interchanges the interior and exterior of the circle, (B) maps circles and lines onto circles and lines, and (C) preserves the angles between two intersecting circles, two intersecting lines, or an intersecting circle and line.

There are, of course, problems with statements of (A) and (B). An inversion in a circle is not defined at the center of the circle, nor does it map any other point of the plane onto the center. As a result, if we restrict to the plane, then an inversion does not map the exterior of the circle *onto* the interior, and will never map a line *onto* a line or *onto* a circle: there will always be a missing point. The only way to clear up this problem so that (A) and (B) become true statements is to invoke the *extended plane* (also called the *conformal plane* or *inversive plane*): the plane extended to include “a point at infinity,” denoted by ∞. Inversion in a circle is then defined to interchange the center with ∞. Yaglom, attempting to stay on an elementary level, downplays the problem by mentioning it not at the start of his discussion but later, and more as a technical remark than as a central issue in the theory (see pages 28–29 and 77). I would prefer the extended plane to be introduced early and more clearly — I think the concept is not that difficult (certainly no harder than many other concepts in the book), and relegating it to a technical issue leads to extra confusion rather than extra clarity.

After establishing the basic properties of inversion Yaglom continues Chapter 1 with several challenging problems (pages 14–22), including Feuerbach’s Theorem (Problem 11) for the nine point circle, and then proves that any two non-intersecting circles (or a circle and a non-intersecting line) can be transformed by an inversion into two concentric circles (part of Theorem 2, page 22). This is followed by several more problems for the reader to solve using this result, including Steiner’s Porism (Problem 16, page 26).

Chapter 2 considers construction problems whose solutions can be simplified by the use of inversions. (Construction problems are featured in all four volumes of *Geometric Transformations*.) Also included are construction problems requiring only a compass and no straightedge, as well as a discussion of the Mohr-Mascheroni Theorem that gives the mild conditions under which a construction can be done just using a compass.

Chapter 3 considers *pencils of circles*, also known as collections of *coaxal circles*. The discussion begins by using the properties of pencils of circles that follow naturally from the properties of inversions established in the previous chapters. The last half of this chapter does not make heavy use of inversions.

Chapter 4 is the final chapter dealing primarily with inversions. It begins with considering what effect inversion has on distance between points. Clearly distance between points is not invariant under inversion; however, Yaglom shows that the (*absolute*)* cross ratio* of four distinct points in the plane is preserved by inversion (page 63), then generalizes this result to the *cross ratio of four circles* (page 65). Yaglom further discusses (page 71) two primary ways in which inversions can be used to establish results: (1) inversions simplify figures associated with relevant problems, which is valuable when the property one desires to prove is invariant under inversion, and (2) inversions produce new theorems from old by significantly changing the configurations under consideration. As always Yaglom supplies problems that illustrate both procedures.

Chapter 4 concludes with the definition of *circular transformation*: a transformation of the plane that maps any circle or line onto a circle or line. (That *Circular Transformations* is the subtitle of this book, and the definition is embedded in a paragraph on page 72, further highlights the problem of lack of an index for this book.) As Yaglom points out later on page 77 the domain for a circular transformation needs to be the *extended* plane if we desire inversions to satisfy this definition. Using the extended plane we obtain that the collection of circular transformations equals the collection of similarities unioned with the collection of inversions composed with similarities (Theorem 2, page 73).

The circular transformations form the group of symmetries for *inversive geometry *on the extended plane, i.e., inversive geometry is the study of properties of figures and transformations on the extended plane that are invariant under circular transformations. This definition does not appear to be mentioned in Yaglom’s text, apparently to keep the exposition at an elementary level. For similar reasons there is no use of complex numbers in the text, and hence no mention that the extended plane can be considered as the *extended complex line* and that the circular transformations correspond to the transformation group generated by the *Möbius transformations* (linear fractional transformations) and complex conjugation. These omissions are not weaknesses; they follow Yaglom’s desire to demonstrate the usefulness of inversive geometry techniques in plane geometry using only basic tools. He does not wish to give a comprehensive description of inversive geometry.

The concluding Chapter 5 considers a more sophisticated type of mapping: *axial circular transformations*, more commonly known as *Laguerre transformations*. Some authors refer to the group of these transformations as the *extended Laguerre group*. The geometric system defined on the plane by the *extended Laguerre group* will be called *Laguerre geometry*. The subgroup of transformations that also preserve the *tangential distance* (Yaglom, page 90) between directed circles is the *restricted Laguerre group*. Some authors use the *restricted Laguerre group* for defining *Laguerre geometry*.

Laguerre transformations are not point mappings. Instead, consider the collection of all directed circles (*cycles*, which include all points considered as non-directed cycles with zero radius) and the collection of all directed lines (*axes*, or *spears*). Yaglom’s definition of an *axial circular transformation* (page 98) is a bijective mapping that takes each axis to an axis and each cycle to a cycle in such a way as to *preserve contact*, i.e., if *S* is a cycle and *l* is a tangent axis (the directions of *S* and *l* must be the same), then the images *S′* and *l′* are also tangent. Any similarity of the plane clearly produces a corresponding axial circular transformation. However, more interesting examples (which do not come from point mappings) are *dilatations* (page 86) and *axial inversions* (pages 105-6). The properties of dilatations and axial inversions are developed in Chapter 5, along with use of these mappings to prove geometric results in the plane (for example, yet another proof of Feuerbach’s Theorem is given in Problem 63, page 93). Yaglom finishes the chapter by classifying an axial circular transformation as either (i) a similarity, (ii) a dilatation followed by a similarity, or (iii) an axial inversion followed by a similarity. (For a *restricted Laguerre transformation* the only change needed in this classification is to replace every occurrence of “similarity” with “isometry.”)

Chapter 5 will be a challenge for any reader who is encountering this sophisticated material for the first time. Yaglom offers a detailed, conversational development, spiraling in on the concepts via manageable and understandable steps. This is a major strength of all four volumes of *Geometric Transformations*. Occasionally this informal style can lead to some confusion, e.g., it takes some thought to fully pull together the definition for axial circular transformations (page 98), but the benefits of Yaglom’s style are far more numerous than the defects.

Apart from the final 113 pages of problem solutions, Volume IV ends with a Supplement focused on the Poincaré disk model for hyperbolic geometry. The *hyperbolic points* in the Poincaré model are the points of the interior of the unit disk, the *hyperbolic lines* are those portions in the open unit disk of Euclidean circles and lines which are perpendicular to the unit circle, and the *hyperbolic motions* are those circular transformations of the plane that map the unit disk onto itself. Each hyperbolic motion can be expressed as a product of inversions (*reflections*) in *hyperbolic lines*. Hence many of the basic results of hyperbolic geometry follow quickly from the results of Chapters I–IV.

Yaglom also gives a short discussion (pages 154–159) of a similar model for elliptic geometry for which the *elliptic points* are all points on the closed unit disk with antipodal boundary points identified, the *elliptic lines* are those portions in the closed unit disk of Euclidean circles and lines passing through antipodal points on the unit circle, and the *elliptic motions* are all those transformations generated by inversions (*reflections*) in *elliptic lines*. The elliptic motions do not preserve the unit disk. However, this model identifies any point in the plane with its image under the inversion of power -1 from the center of the unit circle. With this identification the elliptic motions do map elliptic points to elliptic points.

This model of elliptic geometry is less intuitively appealing than the standard model of the sphere with antipodal points identified (Yaglom indicates these models to be equivalent via stereographic projection on page 167), but it is a satisfying companion to the Poincaré disk model for hyperbolic geometry, much as the spherical model of elliptic geometry is a satisfying companion to the hyperboloid model of hyperbolic geometry. Such pairings illuminate the connections and parallels in structure between the two geometries.

Yaglom ends the Supplement with a comparison of the Poincaré disk model for hyperbolic geometry with the *Klein disk model* as developed in his Supplement to Volume III of *Geometric Transformations*. While the Poincaré model uses the group of circular transformations that preserve the unit disk for its motions, the Klein model uses the group of projective transformations that preserve the unit disk. This model was appropriate for Volume III since the primary focus of that book was on projective transformations. The equivalence of the two models is indicated via the standard method of using stereographic projection (pages 162-3).

(In Volume IV the Klein disk model is said to use the group of *affine* transformations that preserve the unit disk (once on page 160, and four times on page 163) rather than the group of *projective* transformations that the preserve the disk. This is an incorrect description of the transformations, at least given the standard meaning of affine transformation. Volume III correctly uses the term projective, not affine.)

Yaglom’s *Geometric Transformations* is an impressive multi-volume work. Assuming a truly minimum background of basic high school geometry, Yaglom takes his readers on an inspired tour through the group-theoretic, transformational approach to geometry: its philosophy, its list of basic types of transformations, and its use in solving a large collection of interesting and challenging problems of plane geometry. It is a pleasure to now have the fourth and final volume of this series available for English readers. *Geometric Transformations* is a classic of its genre and belongs on the bookshelf of anyone with an interest in the transformational approach to geometry.

William Barker is the Isaac Henry Wing Professor of Mathematics at Bowdoin College. He received his Ph.D. at M.I.T. in 1973, writing a thesis under the guidance of Prof. Sigurdur Helgason in analysis on Lie groups. Barker was subsequently a John Wesley Young Research Instructor at Dartmouth College for two years, joining the Bowdoin faculty in 1975. His most recent work has been an undergraduate geometry textbook, *Continuous Symmetry: From Euclid to Klein*, co-authored with Roger Howe of Yale University. A second volume is currently under development. Barker has also been active with the MAA, taking part in the production of CRAFTY’s *Curriculum Foundations Project* and the CUPM’s *Curriculum Guide 2004.* He can be reached at barker@bowdoin.edu.