Transcendental law of homogeneity

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 is finite and is infinitesimal, then one sets

Similarly,

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]

See also

• Law of continuity
• Adequality

References

Infinitesimals
History	Adequality Leibniz's notation Integral symbol Criticism of non-standard analysis The Analyst The Method of Mechanical Theorems Cavalieri's principle Method of indivisibles
Related branches	Non-standard analysis Non-standard calculus Internal set theory Synthetic differential geometry Constructive non-standard 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
Scientists	Gottfried Wilhelm Leibniz Abraham Robinson Pierre de Fermat Augustin-Louis Cauchy Leonhard Euler
Textbooks	Analyse des Infiniment Petits Elementary Calculus Cours d'Analyse