In the introduction to the first volume we defined geometry as the study of those properties of figures that are not changed by motions; motions were defined as transformations that do not change the distance between any two points of the figure (see pp. 10–11 of Volume One). It follows a t once from this that the most important geometric properties of a figure seem to be the distances between its various points, since the concept of distance between points—the length of a segment—appears to be the most important concept in all geometry. However, if we examine carefully all the theorems of elementary geometry as presented in Kiselyov's text, T then we see that the concept of distance between points hardly figures at all in these theorems. All the theorems on parallel and perpendicular lines (for example, the theorems: “if two parallel lines are cut by a third line, then the corresponding angles are equal” or “from each point not on a given line there is one and only one perpendicular to the given line”), most of the theorems about circles (for example, “through three points not all lying on a straight line one and only one circle can be passed”), many of the theorems about triangles and polygons (for example, “the sum of the angles of a triangle equals a straight angle”, or “the diagonals of a rhombus are perpendicular to each other and bisect the angles of the rhombus”) have nothing whatsoever to do with the concept of distance.