into one of the possible interpretations of nonstandard analysis. Here, the starting point is the "smooth world", a space where all functions are smooth. If this reminds you on Leibniz you will not be astonished to see in proof that in such worlds infinitesimal quantities are possible! In particular, nilpotent infinitesimals are introduced and calculus is shown to work very well with them. In a way this book recovers the notions of "linelets" and "timelets" of Barrow. However, if you really want to understand what is going on in the foundational underground, you have to go deep into model theory. The book ends by just giving a glimpse into this field.