# Transcendental law of homogeneity

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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 (see Leibniz Mathematische Schriften, (1863), edited by C. I. Gerhardt, volume V, pages 377–382). 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.[1] 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 recent study argues that Leibniz's TLH was a precursor of the standard part function over the hyperreals.[2]

## References

1. ^ Bos, Henk J. M. (1974), "Differentials, higher-order differentials and the derivative in the Leibnizian calculus", Archive for History of Exact Sciences, 14: 1–90, doi:10.1007/BF00327456
2. ^ Katz, Mikhail; Sherry, David (2012), "Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond", Erkenntnis, arXiv:, doi:10.1007/s10670-012-9370-y