Law of continuity

The law of continuity is a heuristic principle introduced by Gottfried Leibniz based on earlier work by Nicholas of Cusa and Johannes Kepler. It is the principle that "whatever succeeds for the finite, also succeeds for the infinite".^[1] Kepler used the law of continuity to calculate the area of the circle by representing it as an infinite-sided polygon with infinitesimal sides, and adding the areas of infinitely many triangles with infinitesimal bases. Leibniz used the principle to extend concepts such as arithmetic operations from ordinary numbers to infinitesimals, laying the groundwork for infinitesimal calculus. The transfer principle provides a mathematical implementation of the law of continuity in the context of the hyperreal numbers.

A related law of continuity concerning intersection numbers in geometry was promoted by Jean-Victor Poncelet in his "Traité des propriétés projectives des figures".^[2]^[3]

Leibniz's formulation

Leibniz expressed the law in the following terms in 1701:

In any supposed continuous transition, ending in any terminus, it is permissible to institute a general reasoning, in which the final terminus may also be included (Cum Prodiisset).^[4]

In a 1702 letter to French mathematician Pierre Varignon subtitled “Justification of the Infinitesimal Calculus by that of Ordinary Algebra," Leibniz adequately summed up the true meaning of his law, stating that "the rules of the finite are found to succeed in the infinite."^[5]

The law of continuity became important to Leibniz's justification and conceptualization of the infinitesimal calculus.

References

^ Karin Usadi Katz and Mikhail G. Katz (2011) A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography. Foundations of Science. doi:10.1007/s10699-011-9223-1 See arxiv
^ Poncelet, Jean Victor. Traité des propriétés projectives des figures: T. 1. Ouvrage utile à ceux qui s' occupent des applications de la géométrie descriptive et d'opérations géométriques sur le terrain." (1865), pp. 13–14
^ Fulton, William. Introduction to intersection theory in algebraic geometry. No. 54. American Mathematical Soc., 1984, p. 1
^ Child, J. M. (ed.): The early mathematical manuscripts of Leibniz. Translated from the Latin texts published by Carl Immanuel Gerhardt with critical and historical notes by J. M. Child. Chicago-London: The Open Court Publishing Co., 1920.
^ Leibniz, Gottfried Wilhelm, and Leroy E. Loemker. Philosophical Papers and Letters. 2d ed. Dordrecht: D. Reidel, 1970, p. 544

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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