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 recent study argues that Leibniz's TLH was a precursor of the standard part function over the hyperreals.[3]

See also

• Law of continuity
• Adequality

References

1. Leibniz Mathematische Schriften, (1863), edited by C. I. Gerhardt, volume V, pages 377–382)
2. 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
3. Katz, Mikhail; Sherry, David (2012), "Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond", Erkenntnis, arXiv:1205.0174, doi:10.1007/s10670-012-9370-y