The History of Mathematics: A Brief Course

QUESTIONS AND PROBLEMS 487

equivalent to infinitesimals. He defined a function f(x) to be continuous if the

absolute value of the difference /(x + a) — f(x) "decreases without limit along with

that of Q." He continues:

In other words, the function f(x) remains continuous with respect

to ÷ in a given interval, if an infinitesimal increase in the variable

within this interval always produces an infinitesimal increase in the

function itself.

Certain distinctions that we now make to clarify whether ÷ is a fixed point or

the increase is thought of as occurring at all points simultaneously are not stated

here. In particular, uniform convergence and continuity are assumed but not stated.

Cauchy defined a limit in terms of the "successive values attributed to a variable,"

approaching a fixed value and ultimately differing from it by an arbitrarily small

amount. This definition can be regarded as an informal version of what we now

state precisely with deltas and epsilons, and Cauchy is generally regarded, along

with Weierstrass, as being one of the people who finally made the foundations of

calculus secure. Yet Cauchy's language clearly presumes that infinitesimals are real.

As Laugwitz (1987, p. 272) says:

All attempts to understand Cauchy from a 'rigorous' theory of

real numbers and functions including uniformity concepts have

failed... One advantage of modern theories like the Nonstandard

Analysis of Robinson... [which includes infinitesimals] is that they

provide consistent reconstructions of Cauchy's concepts and results

in a language which sounds very much like Cauchy's.

The secure foundation of modern analysis owes much to Cauchy's treatises.

As Grabiner (1981) says, he applied ancient Greek rigor and modern algebraic

techniques to derive results from analysis. The contributions of other nineteenth-

century mathematicians to this rigor are discussed in Chapter 17.

16.1. Show that the Madhava-Jyeshtadeva formula given at the beginning of the

chapter is equivalent to

16.2. Consider an ellipse with semiaxes a and 6 and a circle of radius b, both circle
and ellipse lying between a pair of parallel lines a distance 26 apart. For every line
between the two lines and parallel to them, show that the portion inside the ellipse
will be a/6 times the portion inside the circle. Use this fact and Cavalieri's principle
to compute the area of the ellipse. This result was given by Kepler.

best remembered. Incidentally, he became a mathematician only after practicing as an engineer
for several years.

Questions and problems

or, letting ÷ — tan È.