In the introduction to NML 8 we defined geometry as a disciplipe concerned with those properties of figures which remain invariant under motions. In the introduction to NML 21 we gave a new definition of geometry as a discipline concerned with those properties of figures which remain invariant under similarities. I t is natural to ask whether or not these definitions are fully equivalent, that is, whether they are different definitions of the same discipline, or whether there exist two different geometries: the one discussed in the introduction to NML 8, and the other discussed in the introduction to NML 21. We shall show that the second alternative is the correct one, that is, that these two geometries are different (though closely related), and in fact, there exist many different geometries. One of the most interesting is the non-Euclidean geometry of Lobachevski-Bolyai, also called hyperbolic geometry; it differs radically from the usual geometry and is discussed in the Supplement at the end of this book.

In the introduction to NML 21 we pointed out that our earlier definition of geometry as the study of those properties of figures which are invariant under motions was inexpedient. We supported this claim as follows: Motions are transformations of the plane that preserve the distance between any two points. However, the number expressing distance depends on the choice of a unit of measurement. Since a geometric proposition cannot depend on the choice of a unit of length, it follows that geometric theorems must refer to ratios of lengths of segments rather than to lengths of segments. Another way of saying this is that in geometry we do not distinguish between similar figures.