Jack Dunn
Newton and Limits Isaac Newton was born on Christmas Day in 1642, the year of Galileo’s death. When he entered Cambridge University in 1661, Newton didn’t know much mathematics, but he learned quickly by reading Euclid and Descartes and by attending the lectures of Isaac Barrow. Cambridge was closed because of the plague in 1665 and 1666, and Newton returned home to reflect on what he had learned. Those two years were amazingly productive for at that time he made four of his major discoveries: (1) his representation of functions as sums of infinite series, including the binomial theorem; (2) his work on differential and integral calculus; (3) his laws of motion and law of universal gravitation and (4) his prism experiments on the nature of light and color. Because of a fear of controversy and criticism, he was reluctant to publish his discoveries and it wasn’t until 1687, at the urging of the astronomer Halley, that Newton published Principia Mathematica. In this work, the greatest scientific treatise ever written, Newton set forth his version of calculus and used it to investigate mechanics, mechanics, fluid dynamics, and wave motion, and to explain the motion of planets and comets. The beginnings of calculus are found in the calculations calcula tions of areas and volumes by ancient Greek scholars such as Eudoxus and Archimedes. Although aspects of the idea of a limit are implicit in their “method of exhaustion”, Eudoxus and Archimedes never explicitly formulated the concept of a limit. Likewise, mathematicians mathematicians such as Cavalieri, Fermat, and Barrow, the immediate precursors of Newton in the development of calculus, did not actually use limits. It was Isaac Newton who was the first to talk explicitly about limits. But Newton acknowledged acknowledged that “If I have seen further than other men, it is because I have stood on the shoulders of giants.” Two of these giants were Pierre Fermat (1601 – 1665) and Newton’s mentor at Cambridge, Isaac Barrow (1630 – 1677). – 1677). Newton was familiar with the methods that these men used to find tangent lines, and their methods played a role in Newton’s eventual formulation of differential calculus. He explained that the main idea behind limits is that quantities “approach nearer than by any given difference.” Newton stated that the limit was the basic concept in calculus, but it was left to later mathematicians like Cauchy to clarify his ideas about limits.
Jack Dunn
Cauchy and Limits After the invention of calculus in the 17 th century, there followed a period of free development of the subject in the 18 th century. Mathematicians like the Bernoulli brothers and Euler were eager to exploit the power of calculus and boldly explore the consequences of this new and wonderful mathematical theory without worrying too much about whether their proofs were completely correct. The 19th century, by contrast, was the Age of Rigor in mathematics. There was a movement to go back to the foundations of the subject – to provide careful definitions and rigorous proofs. At the forefront of this movement was the French mathematician Augustin-Louis Cauchy (1789 – 1857), who started out as a military engineer before becoming a mathematics professor in Paris. Cauchy took Newton’s idea of a limit, which was kept alive in the 18 th century by the French mathematician Jean d’Alembert, and made it more precise. His definition of a limit reads as follows: “When the successive values attributed to a variable approach indefinitely a fixed value so as to end by differing from it by as little as one wishes, this last is called the limit of all the others.” But when Cauchy use d this definition in examples and proofs, he often employed delta-epsilon inequalities similar to the ones in chapter 6. A typical Cauchy proof starts with: “designate by and two very small numbers; …” He used because of the correspondence between epsilon and the French word erreur and because delta corresponds to différence. Later, the German mathematician Karl Weierstrass (1815 – 1897) stated the definition of a limit exactly as in our definition.
