## Casey’s *Treatise on analytical geometry*

*26 December 2008 at 11:27 pm* *
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John Casey’s *Treatise on the analytical geometry of the point, line, circle and conic sections* is a book from the end of the 19th century gathering a wide range of results and constructions in plane geometry known at those times. The *Bibliothèque Nationale de France* gives free access to a digitized version of the book (see there).

The book begins with the definitions of classical coordinate systems: cartesian coordinates, polar coordinates, projective homogeneous coordinates (which were then dubbed trilinear or barycentric). Casey has no taboo concerning points of the field of complex numbers: in particular, many statements make use of cyclic points or zero-radius circles. Much further into the book, the author introduces projections and perspectives, as well as quadratic cones, and harmonic divisions.

Most parts of the book are devoted to triangles and conics. Concerning conics, virtually any result a contemporary mathematician can possibly know about the metric geometry of conics, including foci, auxiliary circles and mostly forgotten theorems of Poncelet, Lemoine, Steiner… are stated, with a lot of exercises. Similarly, most of interesting points, lines or conics associated to a triangle are studied.

The last parts of the book deal with invariants of a pair of conics and the study of their relative position, from a projective or euclidean point of view (you can find there the elegant characterization of foci of a conic using cyclic points), and with duality and reciprocal polars.

Entry filed under: english, geometry, review. Tags: book, conic, geometry, triangle.

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