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| Calculus and Its Origins by David Perkins |
Author
David Perkins examines the extent to which mathematicians and scholars from
Egypt, Persia, and India absorbed and nourished Greek geometry, and details how
the scholars wove their inquiries into a unified theory.
Chapters
cover the story of Archimedes’discovery of the area of a parabolic segment; ibn
Al-Haytham’s calculation of the volume of a revolved area; Jyesthadeva’s
explanation of the infinite series for sine and cosine; Wallis’s deduction
of the link between hyperbolas and
logarithms; Newton’s generalization of the binomial theorem; Leibniz’s
discovery of integration by parts—and much more.
Each
chapter also contains exercises by such mathematical luminaries as Pascal,
Maclaurin, Barrow, Cauchy, and Euler.
Requiring
only a basic knowledge of geometry and algebra—similar triangles, polynomials,
factoring—and a willingness to treat the infinite as metaphor—Calculus & Its Origins is a treasure
of the human intellect, pearls strung together by mathematicians across
cultures and centuries.
MAA Author David Perkins talks about his motivation for writing Calculus and Its Origins and future projects below.
My goal for this book was to fit a
history of calculus into 200 pages that taught calculus while requiring only
basic algebra and geometry of the reader. I did not foresee, when I began
researching, that the pages would fill with such cleverness and beauty from
such a wide variety of times and places.
The MAA editors who
guided me suggested that this work should be a resource for teachers rather
than a textbook for students, so we steered in that direction (although we kept
'exercises' in case the book is assigned in any way). But I
harbor a hope, shared by others in the college community, that the teaching of
calculus will someday soon undergo a transformation that will invite students
to witness the story of the subject's long birth.
I taught calculus
for fifteen semesters at a liberal arts college (Houghton College) before the
weight of failure began to settle on me. To what purpose was I teaching my
students skills—like finding the interval of convergence, estimating areas,
cranking out pages of trig derivatives—all of which I taught well and they
gamely learned; yet when I asked them what calculus is about, they could merely
list the skills they had learned. And if they had turned the question on me,
would my response have mirrored the way I had taught the course? For that
matter, would my response have been anywhere near the truth?
I honestly didn't
know, and it dawned on me that this was because I had no idea what had
motivated the discovery of the subject. Richard Jacobson, head of my department
(and the man to whom I dedicated this book), enthusiastically
gave me permission to teach "Calculus and Its Origins" each spring
semester, which I did three times before I began to write. My teaching
assistants kept asking to be reassigned to the course, and the students who went
on to Calculus 2 did just fine.
I loved learning how thinkers in Greece, the Middle East, and Asia contributed results to the effort that were far more sophisticated than I had imagined. And I loved how so much of what we had all learned in algebra and geometry was being used in support of these contributions.
Finally, I loved being able to demonstrate how scholars ingeniously manipulated the infinitely large and small to mathematically describe the workings of the cosmos: its motions and attractions and flows. I felt like I could finally answer the question, "What is calculus about?"—not merely as a postscript to a skills-based course, but as a theme permeating every class.
I loved learning how thinkers in Greece, the Middle East, and Asia contributed results to the effort that were far more sophisticated than I had imagined. And I loved how so much of what we had all learned in algebra and geometry was being used in support of these contributions.
Finally, I loved being able to demonstrate how scholars ingeniously manipulated the infinitely large and small to mathematically describe the workings of the cosmos: its motions and attractions and flows. I felt like I could finally answer the question, "What is calculus about?"—not merely as a postscript to a skills-based course, but as a theme permeating every class.
I'm currently
working on a book that will require calculus as
it explores the connections between the constants phi, pi, e and i. My plan is
for it to be infused with historical content, and introduce calculus students
to continued fractions, generating functions, and other avenues off the
calculus throughway, while providing a playground in which they can show off
the skills they learned in calculus.
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