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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. 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. 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.

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, S2CID 120779114
3. Katz, Mikhail; Sherry, David (2012), "Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond", Erkenntnis, 78 (3): 571–625, arXiv:1205.0174, doi:10.1007/s10670-012-9370-y, S2CID 254471766

v t e Gottfried Wilhelm Leibniz
Mathematics and philosophy	Alternating series test Best of all possible worlds Calculus controversy Calculus ratiocinator Characteristica universalis Compossibility Difference Dynamism Identity of indiscernibles Individuation Law of continuity Leibniz wheel Leibniz's gap Leibniz's notation Lingua generalis Mathesis universalis Pre-established harmony Plenitude Sufficient reason Salva veritate Theodicy Transcendental law of homogeneity Rationalism Universal science Vis viva Well-founded phenomenon
Works	De Arte Combinatoria (1666) Discourse on Metaphysics (1686) New Essays on Human Understanding (1704) Théodicée (1710) Monadology (1714) Leibniz–Clarke correspondence (1715–1716)
Category

v t e Infinitesimals
History	Adequality Leibniz's notation Integral symbol Criticism of nonstandard analysis The Analyst The Method of Mechanical Theorems Cavalieri's principle
Related branches	Nonstandard analysis Nonstandard calculus Internal set theory Synthetic differential geometry Smooth infinitesimal analysis Constructive nonstandard analysis Infinitesimal strain theory (physics)
Formalizations	Differentials Hyperreal numbers Dual numbers Surreal numbers
Individual concepts	Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of continuity Overspill Microcontinuity Transcendental law of homogeneity
Mathematicians	Gottfried Wilhelm Leibniz Abraham Robinson Pierre de Fermat Augustin-Louis Cauchy Leonhard Euler
Textbooks	Analyse des Infiniment Petits Elementary Calculus Cours d'Analyse

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