For a connected graph G of order $n$, an ordered set $S={u_1,u_2,{cdots},u_k}$ of vertices in G is a linear edge geodetic set of G if for each edge $e=xy$ in G, there exists an index $i$, $1{leq}i$ < $k$ such that e lie on a $u_i-u_{i+1}$ geodesic in G, and a linear edge geodetic set of minimum cardinality is the linear edge geodetic number $leg(G)$ of G. A graph G is called a linear edge geodetic graph if it has a linear edge geodetic set. The linear edge geodetic numbers of certain standard graphs are obtained. Let $g_l(G)$ and $eg(G)$ denote the linear geodetic number and the edge geodetic number, respectively of a graph G. For positive integers $r$, $d$ and $k{geq}2$ with $r$ < $d{leq}2r$, there exists a connected linear edge geodetic graph with rad $G=r$, diam $G=d$, and $g_l(G)=leg(G)=k$. It is shown that for each pair $a$, $b$ of integers with $3{leq}a{leq}b$, there is a connected linear edge geodetic graph G with $eg(G)=a$ and $leg(G)=b$.