In mathematics, the transcendental law of homogeneity (TLH) is a heuristic principle enunciated by

Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...

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 Hendrik Jan Maarten "Henk" Bos (born 17 July 1940, Enschede) is a Dutch historian of mathematics. Career Hendrik was a student of Hans Freudenthal and Jerome Ravetz at Utrecht University and in 1973 wrote a thesis "Differentials, higher order di ...

describes it as the principle to the effect that in a sum involving

infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...

s of different orders, only the lowest-order term must be retained, and the remainder discarded. Thus, if

a

is finite and

dx

is infinitesimal, then one sets :

a+dx=a.

Similarly, :

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 In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every suc ...

over the

hyperreals In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...

See also