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Geometry of Conics

A. V. Akipyan and A. A. Zaslavsky
American Mathematical Society
Publication Date: 
Number of Pages: 
Mathematical World 26
[Reviewed by
Fabio Mainardi
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This book is devoted to the properties of conics that can be formulated and proved using only elementary geometry (by ‘which we mean basically high-school mathematics).

Conics are plane curves of degree two that have been studied since the antiquity (e.g., by Apollonius and Archimedes). Conics are important in astronomy: the orbits of two objects that interact according to Newton’s law of gravitation are conics. From an algebraic, modern, point of view, the classification of conics amounts to a complete classification of quadratic forms (over any field: in the book, however, we deal only with real, ‘physical’ conics). By contrast, the authors adopt a purely geometrical approach; most of the time, they avoid any use of coordinates. Such geometrical proofs are often much, much simpler than their analytical counterparts.

As an example, try to compare geometrical and analytical proofs of the optical properties of conics (described on page 6), which can be stated for the ellipse as follows: Suppose a line L is tangent to an ellipse at a point P; then L is the bisector of the angle FPF’, where F and F’ are the foci of the ellipse.

Most of the theorems proved in the book are classical, yet they cannot be found effortlessly in the literature, and this is, I think, the principal value of the book.

What kind of audience is this book addressed to? As far as I know, its contents are rarely the object of a full course (at least in US and European universities). So I guess its natural target are all those mathematicians (professionals or amateurs) interested in historical aspects, since all the techniques employed in the proofs are those available at the time of Blaise Pascal, author of a famous treatise on conic sections (although Pascal was contemporaneous to Descartes, the discoverer of analytic geometry), and one of the main contributors to the theory.

The book is well written and contains a lot of figures illustrating the theorems and their proofs.

Fabio Mainardi earned a PhD in Mathematics at the University of Paris 13. His research interests are Iwasawa theory, p-adic L-functions and the arithmetic of automorphic forms. He may be reached at

  • Elementary properties of curves of second degree
  • Some results from classical geometry
  • Projective properties of conics
  • Euclidean properties of curves of second degree
  • Solutions to the problems
  • Bibliography
  • Index