The infinite series was a powerful tool employed by Newton and was regarded by him as an essential part of his “method” of analysis. Leibniz preferred a “close form” solution and therefore had little interest in infinite series. Finally, to summarise, both Newton and Leibniz occupy a central position in the history of the calculus. They are considered the inventors because they did something essentially different from and more than the work of their predecessors. Newton and Leibniz discovered methods that were highly important because of their generality5 and could be used to find the solution of many difficult and previously unsolvable problems. What neither did however was establish their methods with the rigour of classical Greek geometry, because both in fact used infinitesimal quantities.
Two very famous problems whose solutions were made possible by the new methods of the differential and integral calculus were the catenary problem and the brachistochrone problem. The catenary is the curve a hanging flexible wire or chain assumes when supported at its ends and acted upon by a uniform gravitational force.8 The first interest in this curve came from Galileo but he wrongly supposed it to be a parabola. The equation was obtained by Leibniz in 1691, and later by Christiaan Huygens and Johann Bernoulli, in response to a challenge by Jakob Bernoulli. In Leibniz’s method of solution he applied the new calculus and found it in a much more direct way.
The brachistochrone problem was a challenging problem posed by Johann Bernoulli in Acta Eruditorum in June 1696. Since Newton and Leibniz were engaged in a bitter priority dispute over the invention of calculus, Johann Bernoulli along with Leibniz deliberately devised the test to find out how much Newton really knew as they were confident that only a person who knows calculus could solve this problem.
The brachistochrone problem, which was addressed, “to acutest mathematicians of the world”, was ‘to find the curve connecting two points, at different heights and not on the same vertical line, along which a body acted upon only by gravity will fall in the shortest time’.9 Bernoulli allowed six months for the solutions but no solutions were received during this period. At the request of Leibniz, the time was publicly extended for a year in order that all contestants should have an equal chance. On 29th of January 1697 the challenge was received by Newton from France and Newton developed the calculus of variations and solved the problem that very night.
The next day (according to his nephew’s memoirs) he sent to Charles Montague, who was then President of the Royal society, his solution. Newton was clearly unimpressed by the challenge set and he wrote in a later response, “I do not love to be dunned and teased by foreigners about mathematical things …” The Royal Society published Newton’s solution anonymously in the Philosophical Transactions of the Royal Society in January 1697.
Bernoulli’s problem was an early example of a class of problems now called Calculus of Variations. These are extremal problems (finding maxima and minima), where the independent variable is not a number but a curve or a function. Calculus allowed Newton and Leibniz to find the solution of many other problems that had been previously unsolvable, including the determination of the laws of motion and the theory of electromagnetism.
Several other types of previously unsolved problems were, the tangent problem (the limit of a sequence), given a position function find the velocity and acceleration, finding the maximum and minimum values of a function, finding the area bounded by a function (the limit of a series), finding the lengths of a curves and finding the volume bounded by a surface. Like any mathematical achievement, the work of Newton and Leibniz on the calculus had to be developed further so that its foundations were secured. This has been achieved by introducing the use of a well defined concept the limit thus creating modern day calculus.