Jesper Lützen is a historian of mathematics and the physical sciences. He is Professor Emeritus at the Department of Mathematical Sciences at the University of Copenhagen, where he has taught since 1989.

• 1: Introduction 2: Prehistory: Recorded and Non-Recorded Impossibilities 3: The First Impossibility Proof: Incommensurability 4: The Classical Problems in Antiquity: Constructions and Positive Theorems 5: The Classical Problems: The Impossibility Question 6: Diorisms and Conclusions about the Greeks and the Medieval Arabs 7: Cube Duplication and Angle Trisection in the 17th and 18th Centuries 8: Circle Quadrature in the 17th Century 9: Circle Quadrature in the 18th Century 10: Impossible Equations Made Possible: The Complex Numbers 11: Euler and the Bridges of Königsberg 12: The Insolvability of the Quintic by Radicals 13: Constructions with Ruler and Compass: The Final Impossibility Proofs 14: Impossible Integrals 15: Impossibility of Proving the Parallel Postulate 16: Hilbert and Impossible Problems 17: Hilbert and Gödel on Axiomatization and Incompleteness 18: Fermat's Last Theorem 19: Impossibility in Physics 20: Arrow's Impossibility Theorem 21: Conclusion

This book is intended as a semi-popular volume: in it, the author eschews mathematical or historical technicalities, instead providing succinct yet rich accounts that neatly convey the main conceptual innovations and transformations at the heart of the episodes discussed therein...The writing is clear and engaging.