These relationships are pretty clear once you see the other distributional metrics. Going further there's skewness and kurtosis for the third and fourth statistical moments, resp.
It doesn't even mention variance (second central moment), let alone skew or kurtosis?
https://stat.ethz.ch/pipermail/r-help/2005-December/083875.h...
"pkg:moments uses the ratio of 4th sample moment to square of second sample moment, while pkg:fBasics uses the variance instead of the second moment and subtracts 3 (for reasons to do with the Normal distribution)."
"The "correct" number for kurtosis depends on your purpose. The number for "kurtosis" that subtracts 3 estimates a "cumulant", which is the standard fourth moment correction weight in an Edgeworth expansion approximation to a distribution. Neither of the numbers described ... compute the "4th sample k statistic", which is "the unique unbiased estimator" for that number (http://mathworld.wolfram.com/k-Statistic.html)."