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Years ago, the handful of peculiar numerical dilogarithmic identities, known since the time of Euler and Landen, gave rise to new discoveries concerning cyclotomic equations and related polylogarithmic ladders. These discoveries were made mostly by the methods of classical analysis, with help from machine computation. About the same time, starting with Bloch's studies on the application of the dilogarithm in algebraic $K$-theory and algebraic geometry, many important discoveries were made in diverse areas. This book seeks to provide a synthesis of these two streams of thought. In addition to an account of ladders and their association with functional equations, the chapters include applications to volume calculations in Lobatchevsky geometry, relations to partition theory, connections with Clausen's function, new functional equations, and applications to $K$-theory and other branches of abstract algebra. This rapidly-expanding field is brought up to date with two appendices, and the book concludes with an extensive bibliography of recent publications. About two-thirds of the material is accessible to mathematicians and scientists in many areas, while the remainder requires more specialized background in abstract algebra. The Evolution Of The Ladder Concept ; Dilogarithmic Ladders / L. Lewin -- Polylogarithmic Ladders ; Ladders In The Trans-kummer Region ; Supernumary Ladders / M. Abouzahra And L. Lewin -- Functional Equations And Ladders / L. Lewin -- Multivariable Polylogarithm Identities / G.a. Ray -- Functional Equations Of Hyperlogarithms ; Kummer-type Functional Equations Of Polylogarithms / G. Wechsung -- The Basic Structure Of Polylogarithmic Equations / Z. Wojtkowiak -- K-theory, Cyclotomic Equations And Clausen's Function / J. Browkin -- Function Theory Of Polylogarithms / S. Bloch -- Partition Identities And Dilogarithm / J.h. Loxton -- The Dilogarithm And Volumes Of Hyperbolic Polytopes / R. Kellerhals -- Introduction To Higher Logarithms / R.m. Hain And R. Macpherson -- Some Miscellaneous Results / L. Lewin. Leonard Lewin, Editor. Includes Bibliographical References (p. 405-408) And Index. The handful of peculiar numerical dilogarithmic identities, known since the time of Euler and Landen, gave rise to the discoveries concerning cyclotomic equations and related polylogarithmic ladders. About the same time, many important discoveries were made in diverse areas. This book intends to provide a synthesis of these two streams of thought.