Les nombres hyperréels

Before the invention of ε&δ notation of modern analysis, philosophers and scientists alike considered infinitesimals and treated them as some kind of peculiar numbers, each of them smaller than 1/n for any natural n. However, the rigorous construction of R does not allow such a number to exist, forcing us to use quite unnatural constructions and roundabout reasoning.
Yet in the middle of XXth century the set of hyperreals was constructed, making rigorous the notion of infinitesimal (and also of number infinitely large), and thus the non standard analysis was created. 
I will follow this construction from the intuition similar to the ideas of Newton and Leibniz to the ordered field of *R, introduce the core of the theory — the transfert principle, and show an example of reasoning in the terms of hyperreals ans supernaturals.