A History of Mathematics- From Mesopotamia to Modernity

192 A History ofMathematics

was achieved in stages through the later nineteenth century by Beltrami, Klein, and Poincaré

among others, by the characteristically modern method of defining ‘models’ for the new geometry.

Of this again, more will follow later.

7. As a result of these developments, it was realized that there was nouniquegeometry, and the
   way was open to the modern understanding of a ‘geometry’ as the study of an axiom-system which
   asserts certain properties of objects called (for example) ‘points’, ‘lines’, etc., without reference to
   what these names may mean. The unique geometry of Euclid has been replaced by a multiplicity
   of geometries, which are equally valid as objects of mathematical study. Because they were the first
   to suggest an alternative to Euclid, Lobachevsky and Bolyai can be seen as the founding fathers of
   this revolution.

Note.The reader who has no idea of what non-Euclidean geometry is, let alone what a model of it

might be, should consider the well-known picture ‘Circle Limit III’ by Moritz Escher (Fig. 3). In this,

which is a picture of non-Euclidean geometry’s version of a plane,

1. the curved lines are to be thought of as straight;


2. all the triangles (and all the fish) are to be considered as having the same size;


3. the bounding circle is ‘at infinity’, and lines which meet there are parallel. The calcula-
   tions underlying the picture are set out by H. S. M. Coxeter at http://www.ams.org/new-in-math/
   circle_limit_iii.pdf.

Fig. 3‘Circle Limit III’ by Moritz Escher.