  The paper is devoted to a detailed self-contained exposition of a part of the
theory of affine planes leading to a construction of affine (or, equivalently,
projective) planes not satisfying the Desarques axiom. It is intended to
complement the introductory expositions of the theory of affine and projective
planes. A novelty of our exposition is a new notation for the ternary operation
in a ternary ring, much more suggestive than the standard one.
