Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus
Judith V. Grabiner, 424 West 7th Street, Claremont, California
91711
The American Mathematical Monthly,
March 1983, Volume 90, Number 3, pp. 185–194.
Student:
The car has a speed of 50 miles an hour. What does that mean?
Teacher:
Given any
there exists a
such that if
then
Student:
How in the world did anybody ever think of such an answer?
P
erhaps this exchange will remind us that the rigorous basis for the calculus is not
at all intuitive—in fact, quite the contrary. The calculus is a subject dealing with
speeds and distances, with tangents and areas—not inequalities. When Newton
and Leibniz invented the calculus in the late seventeenth century, they did not use delta-
epsilon proofs. It took a hundred and fifty years to develop them. This means that it was
probably very hard, and it is no wonder that a modern student finds the rigorous basis of
the calculus difficult. How, then, did the calculus get a rigorous basis in terms of the
algebra of inequalities?
Delta-epsilon proofs are first found in the works of Augustin-Louis Cauchy
(1789–1867). This is not always recognized, since Cauchy gave a purely verbal
definition of limit, which at first glance does not resemble modern definitions: “When
the successively attributed values of the same variable indefinitely approach a fixed
value, so that finally they differ from it by as little as desired, the last is called the
limit
of all the others’’ [
1
]. Cauchy also gave a purely verbal definition of the derivative of
as the limit, when it exists, of the quotient of differences
when
h
goes to zero, a statement much like those that had already been made by Newton,
Leibniz, d’Alembert, Maclaurin, and Euler. But what is significant is that Cauchy
translated such verbal statements into the precise language of inequalities when he
needed them in his proofs. For instance, for the derivative [
2
]:
(1)
Let
be two very small numbers; the first is chosen so that for all numerical
[i.e., absolute] values of
h
less than
and for any value of
x
included [in the
interval of definition], the ratio
will always be greater than
and less than
This one example will be enough to indicate how Cauchy did the calculus, because the
question to be answered in the present paper is not, “how is a rigorous delta-epsilon
proof constructed?’’As Cauchy’s intellectual heirs we all know this. The central
question is, how and why was Cauchy able to put the calculus on a rigorous basis, when
his predecessors were not? The answers to this historical question cannot be found by
reflecting on the logical relations between the concepts, but by looking in detail at the
past and seeing how the existing state of affairs in fact developed from that past.