Yaglom I.M., Shenitzer A. Geometric Transformations IV: Circular Transformations

Mathematical Association of America, 2009. — 285 p.Isaac M. Yaglom, an influential figure in mathematics and mathematics education in the Soviet Union during the 20th 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.)