  A reference frame on a set $M$ is given by a 3-dimensional euclidean space
$E$, a function from $M$ to $E$, a 1-dimensional affine space $A$ and a
function from $M$ to $A$. The definition allows an intuitive and
coordinate-free formulation of Newtonian mechanics, special relativity and
electromagnetism in terms of frames instead of coordinates. In particular, the
postulate of a set of charts on $M$ (an atlas) is replaced by the postulate of
a set of frames. Then the charts can be re-obtained through the choice of
affine coordinates. In addition, units enter the theory as elements of positive
spaces and we obtain a geometric and manifestly unit-independent reformulation.
Each frame allows us to identify spacetime with a 4-dimensional product-space
and we can generalize the definition of differentiable manifolds such that the
frames turn out to form an atlas. Then the only difference between Newtonian
mechanics and special relativity is the assumed type of transition functions -
Galilean transformations or Poincar\'e transformations between affine spaces.
This allows us to highlight the common features and differences in the second
chapter of the paper. For example, we define velocity reciprocity in
mathematical terms and prove the phenomenon for both kind of transformations.
The chapter concludes with a discussion of inertial and accelerated frames. In
the third chapter we restrict our attention to the reformulation of special
relativity and electromagnetism with a strong emphasis on covariance. World
lines are introduced as particular subsets of spacetime, the proper time of
world line is defined as an affine structure on the world line and the
reformulation of electromagnetism is based on the representation of
differential forms w.r.t. a frame.
