Thelaw of continuity is a heuristic principle introduced byGottfried Leibniz based on earlier work byNicholas of Cusa andJohannes 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 toinfinitesimals, laying the groundwork forinfinitesimal calculus. Thetransfer principle provides a mathematical implementation of the law of continuity in the context of thehyperreal numbers.
A related law of continuity concerningintersection numbers ingeometry was promoted byJean-Victor Poncelet in his "Traité des propriétés projectives des figures".[2][3]
Leibniz expressed the law in the following terms in 1701:
In a 1702 letter to French mathematicianPierre 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.
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