The total cross sections for positrons on neon and argon atoms have been measured in the energy ranges 15 eV to 272.5 eV and 25 eV to 300 eV respectively. The cross sections indicate clearly that Born values will not be reached until at least 3 KeV. Interpolating between the measured and the valid Born regions has allowed an application of the sum rule which connects scattering length. Born forward scattering amplitude, and the momentum-integral over the total cross section. This procedure gives scattering lengths as = −0.53 ± 0.15 Bohr radii for neon and as = −2.8 ± 0.7 Bohr radii for argon; the errors include maximum credible uncertainties in the interpolations.