All 950 non-isomorphic simple matroids on 8 elements were constructed in [2] by the use of a computer programme. By using elementary methods, without computer aid, we give another construction for probably the most interesting (when non-isomorphisms are considered) subclass $P$ of 322 rank 4 paving matroids on 8 elements. The class $P$ is partitioned into three disjoint subclasses. The construction of the first two is given in this paper, while the construction of the third subclass of $P$ is given in a sequel paper [1] (these two papers make a whole). We study in greater detail the ways in which the non-isomorphic possibilities arise. Our main tool in the construction of $P$ are three auxiliary classes of graphs (these graphs can be bijected to some of the paving matroids) and some properties of the Steiner system $S(3,4,8)$.