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'. Kepler used The Law of Continuity to calculate the area of the circle by representing the latter 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. A mathematical implementation of the law of continuity is provided by the transfer principle in the context of the hyperreal numbers. 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'. Kepler used The Law of Continuity to calculate the area of the circle by representing the latter 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. A mathematical implementation of the law of continuity is provided by the transfer principle 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'.