Reading Assignment:

Some History of Calculus and its Notations

Obtain a copy of the two articles: The Heroic Century and Calculus Notation. The following questions will guide your reading.

Gårding: The Heroic Century

  1. When was caclulus beging developed? What other important figures (historical, cultural, etc) were alive at the same time as the developers of calculus? (List several.)
  2. Geometrically, 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? By whom?
  3. 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.
  4. 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 mathematics?
  5. The last paragraph on page 28 talks about creation, mathematics and science. How did calculus change people's views about creation? In what way did "Creation [appear] as an even greater miracle than before?"
  6. Near the end of page 29 the author mentions an important advantage to the "abstract point of view.'' What is this advantage?
  7. Archimedes was one of the greatest mathematicians ever, yet although he was working on problems that we would today solve using calculus, he did not discover calculus. What explanations does the author give for this? (There are several.)

Cajori: Calculus Notation

  1. The Calculus notation we use for derivatives and integrals 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.
  2. Repeat the instructions of the last problem, focusing on the notation of
    1. functions,
    2. limits, and
    3. integrals.
  3. Whose names appear prominently in the development of the modern calculus notation? Are their names ones you have heard in other contexts? Where?
  4. What characteristics make a mathematical notation good/desirable?

Thursday, 11-Jan-2001 16:03:12 EST

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