Traditionally (see Johnson et al., 2005, for instance), the (population) standardised skewness and kurtosis can be denoted as
$\sqrt{\beta_1}= \frac{\mu_3}{\sigma^3}\;\;$ and $\;\;\beta_2= \frac{\mu_4}{\sigma^4}$,
respectively, where $\mu_r$ stands for the $r$-th central moment and $\sigma$ denotes the standard deviation of the random variable being considered.
My question is two-fold:
(1) I am curious. Where do those $\beta_1$ and $\beta_2$ actually come from? How/why do they appear? It is just a notation question. How are they originated?
(2) I am confused. Does $\sqrt{\beta_1}$ actually mean $\pm\sqrt{\beta_1}$? I mean, the standadised skewness index $\mu_3/\sigma^3$ can be negative, if I am not wrong. So, why the square root on $\beta_1$? What kind of notation is that? So, this is also a notation question, but in this case I am not sure whether I have correctly understood the meaning of $\sqrt{\beta_1}$.
REFERENCES
Johnson, N. L., Kemp, A. W., & Kotz, S. (2005). Univariate Discrete Distributions (3rd ed.). https://doi.org/10.1002/0471715816