PRINCIPLES OF ITERATIONAL CALCULUS. WRITTEN BY  L. E. B ¨ OTTCHER. TRANSLATION FROM THE POLISH BY MA LGORZATA STAWISKA Part One. Iterational calculus as a special chapter of the theory of one-parameter continuous transformation groups 1. General notions. §1 Generalization of the notions of a variable and a function. In order to achieve suitable generality of the notion of iteration, we will generalize the notions of a variable and a function. The location of an arbitrary point in an n-dimensional space is pre- cisely determined if we know values of all of its n coordinates (x 1 , ..., x n ). Therefore the tuple (x 1 , ..., x n ), as a representation of the location of a point in an n-dimensional space, will be regarded as a generalized number. If the point X corresponding to the tuple (x 1 , ..., x n ) re- mains at rest with respect to the coordinate system, then the “number” (x 1 , ..., x n ) has the nature of a “constant number”; if however the point X changes perpetually its location with respect to the coordinate sys- tem, then the “number” (x 1 , ..., x n ) has the nature of a “variable num- ber”. A system of equations x ′ i = f (x 1 , ..., x n ); (i =1, 2, ..., n) states that the n-dimensional space is subject to a transformation such that the point X changes its location and takes a new location X ′ , or, equiv- alently, the tuple (x 1 , ..., x n ) is transformed into the tuple (x ′ 1 , ..., x ′ n ). The equations x ′ i = f (x 1 , ..., x n ); (i =1, 2, ..., n) define a certain transformation (x ′ 1 , ..., x ′ n )= T (x 1 , ..., x n ), which as a result gives us the tuple (x ′ 1 , ..., x ′ n ) as some function of the tuple (x 1 , ..., x n ). The tuple (x 1 , ..., x n ) occurring as an “independent variable” is called an “argument”, and the tuple (x 1 , ..., x n ) occurring as a “dependent vari- able” is called a “function”. The generalized notion of a function should be understood as follows: the tuple (x ′ 1 , ..., x ′ n ) is called a function of the tuple (x 1 , ..., x n ) if by some transformation T (x 1 , ..., x n ) the tu- ple (x 1 , ..., x n ) is transformed into the tuple (x ′ 1 , ..., x ′ n ), i.e., if the 1