  The present paper contributes to the ongoing programme of quantification of
isomorphic Banach space theory focusing on Pe{\l}czy\'nski's classical work on
dual Banach spaces containing $L_{1}$ ($=L_{1}[0,1]$) and the Hagler--Stegall
characterisation of dual spaces containing complemented copies of $L_{1}$. We
prove the following quantitative version of the Hagler--Stegall theorem
asserting that for a Banach space $X$ the following statements are equivalent:
  $\bullet$ $X$ contains almost isometric copies of $(\bigoplus_{n=1}^{\infty}
\ell_{\infty}^{n})_{\ell_1}$,
  $\bullet$ for all $\varepsilon>0$, $X^{*}$ contains a
$(1+\varepsilon)$-complemented, $(1+\varepsilon)$-isomorphic copy of $L_{1}$,
  $\bullet$ for all $\varepsilon>0$, $X^{*}$ contains a
$(1+\varepsilon)$-complemented, $(1+\varepsilon)$-isomorphic copy of
$C[0,1]^{*}$.
  Moreover, if $X$ is separable, one may add the following assertion:
  $\bullet$ for all $\varepsilon>0$, there exists a $(1+\varepsilon)$-quotient
map $T\colon X\rightarrow C(\Delta)$ so that $T^{*}[C(\Delta)^{*}]$ is
$(1+\varepsilon)$-complemented in $X^{*}$, where $\Delta$ is the Cantor set.
