Transcendental law of homogeneity

In mathematics, the transcendental law of homogeneity (TLH) is a heuristic principle enunciated by Gottfried Wilhelm Leibniz most clearly in a 1710 text entitled Symbolismus memorabilis calculi algebraici et infinitesimalis in comparatione potentiarum et differentiarum, et de lege homogeneorum transcendentali.^[1] Henk J. M. Bos describes it as the principle to the effect that in a sum involving infinitesimals of different orders, only the lowest-order term must be retained, and the remainder discarded.^[2] Thus, if ${\displaystyle a}$ is finite and ${\displaystyle dx}$ is infinitesimal, then one sets

${\displaystyle a+dx=a.}$

Similarly,

${\displaystyle u\,dv+v\,du+du\,dv=u\,dv+v\,du,}$

where the higher-order term du dv is discarded in accordance with the TLH. A 2012 study argues that Leibniz's TLH was a precursor of the standard part function over the hyperreals.^[3]