The Law of Continuity is a heuristic principle introduced by Leibniz based on earlier work by Nicholas of Cusa

Nicholas of Cusa

Nicholas of Kues , also referred to as Nicolaus Cusanus and Nicholas of Cusa, was a cardinal of the Catholic Church from Germany , a philosopher, theologian, jurist, mathematician, and an astronomer. He is widely considered one of the great geniuses and polymaths of the 15th century...

 and Johannes Kepler

Johannes Kepler

Johannes Kepler was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican...

. It is the principle that "whatever succeeds for the finite, also succeeds for the infinite". Kepler used it 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 infinitesimal

Infinitesimal

Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...

s, laying the groundwork for infinitesimal calculus

Infinitesimal calculus

Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...

. A mathematical implementation of the law of continuity is provided by the transfer principle

Transfer principle

In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure...

 in the context of the hyperreal number

Hyperreal number

The system of hyperreal numbers represents a rigorous method of treating the infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form1 + 1 + \cdots + 1. \, Such a number is...

s.