It took two millennia to demonstrate that some famous Greek problems cannot be solved using only a straight edge and compass. The pursuit of squaring the circle, trisecting the angle, and duplicating the cube engaged many of history's greatest mathematical minds, including Archimedes, Euclid, Newton, Fermat, Gauss, and Descartes. This exploration not only highlights their dedication to these challenges but also showcases the mathematical innovations that emerged from them. The journey through these problems can be divided into three distinct periods: from the Greeks to 17th-century calculus and analytic geometry, and finally to 19th-century advancements in irrational and transcendental numbers. The author, a mathematics teacher, dedicates a chapter to each problem and includes additional chapters on complex numbers and criteria for constructibility. With commentary and problem sets following each chapter, amateur puzzlers are guided through these complexities. A basic understanding of calculus will aid readers, with full solutions provided at the end. Students and enthusiasts eager to engage with these historical challenges and the evolution of modern mathematics will find this exploration both enlightening and stimulating. Discover the pivotal moments that shaped mathematical thought, including Gauss's inspiration from a vision of a 17-sided polygon and the significance of the equation e[pi][i] = -1.
