Reading Assignment #1
Obtain a copy of the two articles: “The Heroic Century”
and “Calculus Notation”. Read these articles and
answer the following questions (which are different, for
the most part, than the ones at the end of the articles
themselves.
Gårding: The Heroic Century
- If differential calculus is concerned with finding tangents
to curves and integral calculus with finding areas under
curves, which one of these seems to be the problem being
investigated first historically.
- Along with finding areas and tangents, what are some of
the other problems that the mathematicians mentioned
in the reading were studying. Wherever possible, be
specific about who was studying what.
- Gårding talks about Archimedes' method of carefully
proving all of his mathematical assertions, and how
some investigators of the 17th Century got away from this.
If there were not the systematic theory in place to
justify the methods of calculus, why did so many great
minds of the time embrace the subject?
In what ways does it seem that mathematics is like the
other sciences? What role does rigorous proof play in
the subject?
- One of the greatest tools in mathematics is a creative
mind — in particular, the ability to look at a
difficult problem in a way different than conventional
methods suggest, and find something of value in your
new viewpoint that makes a solution possible. While
Archimides in the 17th Century was looked at as one
of the greatest mathematicians (and still is looked
at that way), his efforts on calculus-type problems
were not sufficient to reveal to him the crucial
insights later seen by Newton and Leibniz. What does
Gårding think was the cause of this? Can you offer
reasons that he does not mention?
Cajori: Calculus Notation
- The Calculus notation we use for derivatives (and
integrals, if you've seen such things) has been in use
for a long time. What is clear from this article is
that, in the early days, it was a struggle to
decide upon what types of notations people should use.
Focusing just on derivative notation, explain in detail
(give examples) various old and modern ways of representing
derivative expressions. Give some reasons why you think
the old/modern notations are good or bad, avoiding the
urge to declare something good just because you are
accustomed to it.
- Repeat the instructions of the last problem,
focusing on the notation of
functions and limits (and for integrals,
too, if you like).
- Whose names appear prominently in the development
of the modern Calculus notation? Are their names ones
you have heard in other contexts? Where?
- Do Exercise 3 at the end of this reading.
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Thomas L. Scofield
Department of Mathematics and Statistics
Calvin College
Last Modified:
Monday, 26-Jul-2004 13:10:38 EDT