= Solution
Rearrange the given <torsion balance> equilibrium relation as
$$
\theta=\frac{GmML}{\kappa}\frac1{r^2}.
$$
A useful choice, preserving equal measurement uncertainties on the vertical axis, is
$$
\boxed{y=\theta,\qquad x=r^{-2},\qquad c=\frac{GmML}{\kappa}.}
$$
With <angle> measured in radians, $y$ is dimensionless and $x$ has units $\mathrm{m^{-2}}$, so $c$ has units $\mathrm{m^2}$, as required. The four $x$ values, in the order of the original table, are
$$
(3906.25,\ 3086.41975,\ 2066.11570,\ 976.56250)\,\mathrm{m^{-2}}.
$$
The corresponding $y$ values are $(0.066,0.055,0.044,0.025)$, each with <standard uncertainty> $\sigma_y=0.002$. The uncertainty of $x$ is negligible by assumption, making <linear regression through the origin> appropriate for the proposed model.
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