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Nikolai I. Lobachevsky
European Mathematical Society
Publication Date: 
Number of Pages: 
Heritage of European Mathematics
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Charles Ashbacher
, on

Mathematics and religion are not often discussed together, but there certainly seems to have been something quasi-religious about mathematicians’ attachment to Euclidean geometry. When non-Euclidean geometry first made an appearance, many still considered Euclidean geometry “the true geometry” and they fought back against the new ideas by unleashing a flurry of attempts to prove Euclid’s fifth postulate. In their minds, only Euclidean geometry could be used to describe the physical world. They all failed; non-Euclidean geometry could not be denied in the end. The section “On Hyperbolic Geometry and Its Reception” details much of the controversy that surrounded the first results that established hyperbolic geometry.

The main section of the book consists of three versions (English, French and Russian) of “Pangeometry”, as originally written by Lobachevsky in 1855. There is something mystical about reading the thoughts of groundbreaking mathematicians as they first put their revolutionary thoughts down on paper. You see them in their original form, before decades of polishing, improving, verifying and correcting have been done. It allows you to experience their genius, as they broke with the past and set mathematics on a new course from which it will never veer. This is especially true for hyperbolic geometry; it was a radical break from the past and required a significant change in thought patterns.

The great Gauss had as his motto “few but ripe”, referring to his mathematical publications. That could be modified to say, “it is not the number, it is the quality and significance of the publications that matter.” That is exceptionally true in the case of Lobachevsky, for his “Pangeometry” changed our very perception of the universe. In many ways you will be surprised at the simplicity of his ideas.



Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, teaching college classes and co-editing The Journal of Recreational Mathematics. In his spare time, he reads about these things and helps his daughter in her lawn care business.

  • I. Pangeometry
  • II. Lobachevsky's biography
  • III. A commentary on Lobachevsky's Pangeometry
  • Bibliography