The purpose of this publication as a pedagogical contribution is to illustrate the uniqueness of the method Euler used to determine the unsteady flow of blood in the circulatory system. In his presentation, he accounts for the tapering of the vessels and their elastic behavior. The limitations of this study are based upon the extent of the development of the mathematics of his era. This work may be the first mathematical treatment of circulatory physiology and haemodynamics. Euler perhaps could very well be called the Father of Haemodynamics.