On the first page of the high school geometry text by A. P. Kiselyov, immediately after the definitions of point, line, surface, body, and the statement “a collection of points, lines, surfaces or bodies, placed in space in the usual manner, is called a geometric figure”, the following definition of geometry is given: “Geometry is the science that studies the properties of geometric figures.” Thus one has the impression that the question posed in the title to this introduction has already been answered in the high school geometry texts, and that it is not necessary to concern oneself with it further.

But this impression of the simple nature of the problem is mistaken. Kiselyov's definition cannot be called false; however, it is somewhat incomplete. The word “property” has a very general character, and by no means all properties of figures are studied in geometry. Thus, for example, it is of no importance whatever in geometry whether a triangle is drawn on white paper or on the blackboard; the color of the triangle is not a subject of study in geometry. It is true, one might answer, that geometry studies properties of geometric figures in the sense of the definition above, and that color is a property of the paper on which the figure is drawn, and is not a property of the figure itself. However, this answer may still leave a certain feeling of dissatisfaction; in order to carry greater conviction one would like to be able to quote a precise “mathematical” definition of exactly which properties of figures are studied in geometry, and such a definition is lacking.