This book is the concluding Part IV of *Geometric Transformations*, but it can be studied independently of Parts I, II, and III, which appeared in this series as Volumes 8, 21, and 24. Part I treats the geometry of rigid motions of the plane (isometries); Part II treats the geometry of shape-preserving transformations of the plane (similarities); Part III treats the geometry of transformations of the plane that map lines to lines (affine and projective transformations) and introduces the Klein model of non-Euclidean geometry.

The present Part IV develops the geometry of transformations of the plane that map circles to circles (conformal or anallagmatic geometry). The notion of inversion, or reflection in a circle, is the key tool employed. Applications include ruler-and-compass constructions and the Poincaré model of hyperbolic geometry.

### Table of Contents

1. Reflection in a circle (inversion)

2. Applications of inversions to the solution of constructions

3. Pencils of circles. The radical axis of two circles

4. Inversion (concluding section)

5. Axial circular transformations

Supplement

Solutions

About the Author

### About the Author

**Isaac Moisevitch Yaglom** was born on March 6, 1921 in the city of Kharkov. He graduated from Sverdlovsk University in 1942 and received his Candidate’s Degree (the equivalent of an American Ph.D.) from Moscow State University in 1945. He received the D.Sc. degree in 1965. An influential figure in mathematics education in the Soviet Union, he was the author of many scientific and expository publications. In addition to Geometric Transformations, English translations of his books include *Convex Figures* (Holt, Rinehart and Winston, 1961, written jointly with V.G. Boltyanskii), *Challenging Mathematical Problems with Elementary Solutions* (Holden-Day, 1964, written jointly with his twin brother Akiva M. Yaglom), *Complex Numbers in Geometry* (Academic Press, 1968), *A Simple Non-Euclidean Geometry and Its Physical Basis* (Springer, 1979), *Probability and Information* (Reidel, 1983, written jointly with Akiva), *Mathematical Structures and Mathematical Modelling* (Gordon and Breach, 1986), and *Felix Klein and Sophus Lie* (Birkäuser, 1988). Professor Yaglom died April 17, 1988 in Moscow.

**Abe Shenitzer**, the translator of the present volume, is professor emeritus of mathematics at York University in Toronto. Born in Warsaw in 1921, he attended Brooklyn College and received his Ph.D. from New York University in 1954.

### MAA Review

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. Continued...