In this last lecture “A fascinating tool for computations in Early Antiquity: the Anthyphairesis”, I will present the method called by ancient Greek mathematicians ‘anthyphairesis’ i.e. ‘alternate subtraction’. I will try to understand the origins of this simple but sophisticated algorithm which is the ancestor of what modern mathematicians call ‘continuous fractions’. I will show how it was, or at least could had been used to approximate some fractions in the commensurable case. Finally I will consider whether, as claimed by many modern mathematicians and historians of mathematics, there is any possibility the ‘anthyphairesis’ was a tool at the origins of the discovery of the irrationality.