Using standard analysis only, we present an extension $\ER$ of the real field containing nilpotent infinitesimals. On the one hand we want to present a very simple setting to formalize infinitesimal methods in Diffe\-ren\-tial Geometry, Analysis and Physics. On the other hand we want to show that these infinitesimals may be also useful in infinite dimensional Differential Geometry, e.g. to study spaces of mappings. We define a full embedding of the category $\Man$ of finite dimensional $\Cn$ manifolds in a cartesian closed category. In it we have a functor $\ext(-)$ which extends these spaces adding new infinitesimal points and with values in another full cartesian closed embedding of $\Man$. We present a first development of Differential Geometry using these infinitesimals.