To the Instructor
When Newton and Leibnii3 invented the calculus in the late sev-
enteenth century, they did not use delta-epsilon proofs. It took a
hundred and fifty years to develop them.... [It} is no wonder
that a modern student finds the rigorous basis of calculus difficult.
Delta-epsilon proofs are first found in the works of Augustin-
Louis Cauchy {1789- 1867).
Cauchy, followed by Riemann and Weierstrass,^4 gave the calcu-
lus a rigorous basis.- Judith V. Grabiner [56]
Many reasons can be given to explain why the standard introductory real anal-
ysis course has come to assume its present form, with its reliance upon rigorous
proofs based on the r::-6 definition of limit. As you know, analysis had a bril-
liant history of remarkable achievements for almost two centuries without the
benefit of a rigorous foundation. Mathematicians such as the Bernoullis, Euler,
Lagrange, Laplace, Fourier, and Gauss^5 made many brilliant discoveries with-
out seeming to need rigor as defined by today's standards. The development
of the theory of Fourier series is often cited as one of the principal motivators
of the change in attitude toward rigor. Careful and critical study of Fourier
series required a more rigorous understanding of such basic concepts as func-
tion, limit, continuity, and convergence. It was also in this connection that a
proper definition of the definite integral was needed, which led to a rigorous
development of the Riemann integral, and ultimately to the Lebesgue integral
and still other generalized integrals. Rigor thus came to be seen as essential to
the core of analysis, not just as an afterthought.
- Sir Isaac Newton (1642- 1727) and Gottfried Wilhelm Leibniz (1646- 1716).
- Karl Weierstrass (1815- 1897) and Bernard Riemann (1826- 1866).
- J acob and Johann Bernoulli (1654- 1705 and 1667 - 1748), Leonard Euler (1707- 1783),
Joseph Louis Lagrange (1736- 1863), Pierre Simon de Laplace (1749- 1827), Joseph Fourier
(1768- 1830), and Carl Friedrich Gauss (1777- 1855).
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