The lengthening shadow

The Lengthening Shadow: The Story of Related Rates
Bill Austin University of Tennessee at Martin, Martin, TN 38238
Don Barry Phillips Academy, Andover, MA 01810
David Berman University of New Orleans, New Orleans, LA 70148
Mathematics Magazine, February 2000, Volume 73, Number 1, pp. 3–12.
Introduction

A

boy is walking away from a lamppost. How fast is his shadow moving? A ladder
isresting against a wall. If the base is moved out from the wall, how fast is the
top of the ladder moving down the wall?

Such “related rates problems” are old chestnuts of introductory calculus, used both to
show the derivative as a rate of change and to illustrate implicit differentiation. Now that
some “reform” texts [4, 14] have broken the tradition of devoting a section to related
rates,it is of interest to note that these problems originated in calculus reform
movements of the 19th century.

Ritchie, related rates, and calculus reform
Related rates problems as we know them date back at least to 1836, when the Rev.
William Ritchie (1790–1837), professor of Natural Philosophy at London University
1832–1837, and the predecessor of J. J. Sylvester in that position, publishedPrinciples
of the Differential and Integral Calculus. His text [21, p. 47] included such problems as:
If a halfpenny be placed on a hot shovel, so as to expand uniformly, at what rate
is its surface increasing when the diameter is passing the limit of 1 inch and 1/10,
the diameter being supposed to increase uniformly at the rate of .01 of an inch per
second?

This related rates problem wasno mere practical application; it was central to Ritchie’s
reform-minded pedagogical approach to calculus. He sought to simplify the presentation
of calculus so that the subject would be more accessible to the ordinary, non-university
student whose background might include only “the elements of Geometry and the
principles of Algebra as far as the end of quadratic equations” [21, p. v]. Ritchiehoped
to rectify what he saw as a deplorable state of affairs:
The Fluxionary or Differential and Integral Calculus has within these few years
become almost entirely a science of symbols and mere algebraic formulae, with
scarcely any illustration or practical application. Clothed as it is in a
transcendental dress, the ordinary student is afraid to approach it; and even many
of those whoseresources allow them to repair to the Universities do not appear to
derive all the advantages which might be expected from the study of this
interesting branch of mathematical science.

Ritchie’s own background was not that of the typical mathematics professor. He had
trained for the ministry, but after leaving the church, he attended scientific lectures in
Paris, and “soon acquired greatskill in devising and performing experiments in natural
philosophy. He became known to Sir John Herschel, and through him [Ritchie]
communicated [papers] to the Royal Society” [24, p. 1212]. This led to his appointment
as the professor of natural philosophy at London University in 1832.

To make calculus accessible, Ritchie planned to follow the “same process of thought by
which we arrive atactual discovery, namely, by proceeding step by step from the
simplest particular examples till the principle unfolds itself in all its generality.”
[21, p. vii; italics in original]
Drawing upon Newton, Ritchie takes the change in a magnitude over time as the
fundamental explanatory concept from which he creates concrete, familiar examples
illustrating the ideas of calculus. He begins with anintuitive introduction to limits
through familiar ideas such as these; (i) the circle is the limit of inscribed regular
polygons with increasing numbers of sides; (ii) 1 9 is the limit of
1 10 1 100 1 1000 . . . ; (iii) 1 2x is the limit of h 2xh h2 as h approaches
0. Then—crucial to his pedagogy—he uses an expanding square to introduce both the
idea of a function and the fact that a...