The law of continuity is a heuristic principle introduced by

Gottfried 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 mathem ...

based on earlier work by

Nicholas of Cusa Nicholas of Cusa (1401 – 11 August 1464), also referred to as Nicholas of Kues and Nicolaus Cusanus (), was a German Catholic cardinal, philosopher, theologian, jurist, mathematician, and astronomer. One of the first German proponents of Re ...

and

Johannes Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...

. 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 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

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 re ...

s, laying the groundwork for

infinitesimal calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of ari ...

. The transfer principle provides a mathematical implementation of the law of continuity in the context of the

hyperreal number 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 ...

s. A related law of continuity concerning

intersection number In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for t ...

s in

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

was promoted by

Jean-Victor Poncelet Jean-Victor Poncelet (; 1 July 1788 – 22 December 1867) was a French engineer and mathematician who served most notably as the Commanding General of the École Polytechnique. He is considered a reviver of projective geometry, and his work '' ...

in his "Traité des propriétés projectives des figures".

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''). In a 1702 letter to French mathematician

Pierre Varignon Pierre Varignon (1654 – 23 December 1722) was a French mathematician. He was educated at the Jesuit College and the University of Caen, where he received his M.A. in 1682. He took Holy Orders the following year. Varignon gained his first ...

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."Leibniz, Gottfried Wilhelm, and Leroy E. Loemker. Philosophical Papers and Letters. 2d ed. Dordrecht: D. Reidel, 1970, p. 544 The law of continuity became important to Leibniz's justification and conceptualization of the infinitesimal calculus.

References

{{Infinitesimals Nonstandard analysis Gottfried Wilhelm Leibniz Infinity History of calculus Mathematics of infinitesimals

Leibniz's formulation

See also

References