  We study and compare the webs $\boldsymbol{\mathcal W}_{{\rm dP}_d}$ defined
by the conic fibrations on a given smooth del Pezzo surface ${\rm dP}_d$ of
degree $d$ for $d=4$ and $d=5$. In a previous paper, we proved that for any
positive $d\leq 6$, the web by conics $\boldsymbol{\mathcal W}_{{\rm dP}_d}$
carries a particular abelian relation ${\bf HLog}_d$, whose components all are
weight $7-d$ antisymmetric hyperlogarithms. The web $\boldsymbol{\mathcal
W}_{{\rm dP}_5}$ is a geometric model of the exceptional Bol's web and the
relation ${\bf HLog}_5$ corresponds to the famous `Abel's identity'
$(\boldsymbol{{\mathcal A}b})$ of the dilogarithm. Bol's web together with
$(\boldsymbol{{\mathcal A}b})$ enjoy several remarkable properties of different
kinds. We show that almost all of them admit natural generalizations to the
pair $\big( \boldsymbol{\mathcal W}_{{\rm dP}_4}, {\bf HLog}_4\big)$.
