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Later editors have interpolated Euclid’s implicit axiomatic assumptions in the list of formal axioms.įor example, in the first construction of Book 1, Euclid uses a premise that was neither postulated nor proved: that two circles with centers at the distance of their radius will intersect in two points. His proofs often invoke axiomatic notions, which were not originally presented in his list of axioms. His current mathematical interests are in modular arithmetic and cryptography.While Euclid’s list of axioms in the “Elements” is not exhaustive, it represents the most important principles. He hopes to pursue mathematics in his post-secondary education. Jack Chen ( ) is a high school student who enjoys mathematics. Overall, the book is a fine resource for readers who are new to geometry as well as those who want to sharpen their geometric skills. Given that the book is aimed at the pre-university and advanced high-school levels, some parts of the book, especially chapter 5, may prove difficult.
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The solutions, much like the book itself, are thorough and easy to follow. Many of the exercises in the book have hints, and the exercises without hints aren’t difficult enough to merit one. The exercises cultivate appreciation of geometry and encourage mathematical play, but additional drill exercises would have been nice to help readers master important concepts. Sasane’s Plain Plane Geometry covers the fundamentals of planar geometry while also introducing additional topics. Aspiring geometers may feel overwhelmed by the challenging final chapter, but it’s an excellent chapter nonetheless. The theme is circles, including topics such as tangent lines, Simson’s line, and Ptolemy’s theorem. The difficulty again increases in chapter 5. There are several memorable exercises, including one where readers can estimate the radius of the Earth with similar triangles. The focus is now on similar triangles, and the chapter includes topics such as similarity rules and Menelaus’ theorem. These digressions and applications show off the beauty of geometry and are quite enjoyable. For example, Kepler’s second law is discussed and proved using nothing more than basic geometry. In this and latter chapters, there are several digressions into the history of geometry along with multiple real-world applications. Quadrilaterals become the theme in chapter 3, along with topics such as the midpoint theorem, area, and Ceva’s theorem. Chapter 2 is on congruent triangles, and includes topics such as congruency rules, geometric constructions, and concurrency. The fundamentals of geometry, from points to polygons, are rigorously defined. The book follows the “Definition-Theorem-Proof-Exercise” format and begins with an introductory chapter on geometric figures. More importantly, it does communicate the beauty and charm of geometry. I believe that Sasane’s book is an excellent gateway into planar geometry: it is accessible, systematic, and thorough.
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The aim of this book is to cover the basics of the wonderful subject of Planar Geometry, at high school level, requiring no prerequisites beyond arithmetic, and hopefully to convey the sense of joy which I had when I was taught geometry.Ī review of Plain Plane Geometry must necessarily be based upon the degree to which Sasane fulfills this aim.