leibniz

Mathematical Competitions

Before Leibniz's journey to Italy the calculus had been developed to a point where Leibniz was able to differentiate and integrate – albeit very often using series expansions – the greater part of the functions then known. He had succeeded too in solving simpler types of differential equations. In 1687 Leibniz embarked on a public challenge to the Cartesians whom he called on to find the curve of uniform descent of a body in the earth's gravitational field (isochrone).

A little later Leibniz extended the formulation of this problem such that the descending body was to approach a given point at a constant speed (isochrona paracentrica). The solution of this extended problem however was deferred for five years.

In the year 1690 Jacob Bernoulli raised the question – first posed by Galileo but still unanswered – as to the form of the curve taken by a flexible non-ductile chain fixed at two points of equal altitude and suspended in the earth's gravitational field (catenary). Since in the eyes of the Italians the reputation of their indeed most famous scholar had been tainted, Viviani responded with the formulation of the Florentine problem, namely to cut out from a hemisphere four similarly arranged windows such that an exact quadrature of the remaining surface be possible. Following the immediate solution of this problem by the mathematicians north of the Alps, Johann Bernoulli challenged the learned community to determine all curves where an intercept on the axis at the vertex by a tangent at a point be in a constant relation (M : N) to the length this tangent (Bernoulli problem).

Finally, the zenith of this contest was reached in 1696 with the formulation, likewise by Johann Bernoulli, of the Brachystochrone problem, which required the determination of the curve along which a body in the earth's gravitational field moves in the shortest possible time from a point A to an inferior point B. With the formulation of this problem was created the new sub-discipline, the calculus of variations, which in turn was rapidly promoted in the time that followed through the efforts for an adequate solution of the longstanding isoperimetric problem. – In the efforts to solve the aforementioned challenge questions all the leading mathematicians of Europe were involved and Leibniz made a major contribution to the solution of each these problems.

Further reading:   G.W. Leibniz. La naissance du calcul différentiel. Ed M. Parmentier (Paris, 1989).

back to: Leibniz´ life and work

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