  Optical precision spectroscopy of isotope shifts can be used to test for new
forces beyond the Standard Model, and to determine basic properties of atomic
nuclei. We measure isotope shifts on the highly forbidden ${}^2S_{1/2}
\rightarrow {}^2F_{7/2}$ octupole transition of trapped
$^{168,170,172,174,176}$Yb ions. When combined with previous measurements in
Yb$^+$ and very recent measurements in Yb, the data reveal a King plot
nonlinearity of up to 240$\sigma$. The trends exhibited by experimental data
are explained by nuclear density functional theory calculations with the Fayans
functional. We also find, with 4.3$\sigma$ confidence, that there is a second
distinct source of nonlinearity, and discuss its possible origin.
