  The circumference of a graph is the length of its longest cycles. Jackson
established a conjecture of Bondy by showing that the circumference of a
3-connected cubic graph of order $n$ is $\Omega(n^{0.694})$. Bilinski {\it et
al.} improved this lower bound to $\Omega(n^{0.753})$ by studying large
Eulerian subgraphs in 3-edge-connected graphs. In this paper, we further
improve this lower bound to $\Omega(n^{0.8})$. This is done by considering
certain 2-connected cubic graphs, finding cycles through two given edges, and
distinguishing the cases whether or not these edges are adjacent.
