  We consider a classial case of irrational integrals containing a square root
of a quadratic polynomial. It is well known that they can be expressed in terms
of elementary functions by one of three Euler's substitutions. It is less known
that the Euler substittutions have a beautiful geometric interpretation. In the
framework of this interpretation one can see that the number 3 is not the most
suitable. We show that it is natural to introduce the fourth Euler
substitution. By the way, it is not clear who was the first to attribute these
three substitutions to Euler. In his original treatise Leonhard Euler uses two
substitutions which are sufficient to cover all cases.
