Cut The Knot!

An interactive column using Java applets
by Alex Bogomolny

Skimming A Century of Calculus

December 2000

There are many conversations, committee meetings, etc., today about the modernization of the undergraduate calculus course; but all too often the attack on the problem falls short of being comprehensive. So begins a 1958 article Bringing Calculus Up-to-date (AMM, 65 (1958), pp. 81-90) by M. E. Munroe. Too young at the time to be interested in Calculus, I first ran into the article while browsing the first volume of the collection A Century of Calculus. It's a remarkable two-volume set -- a small library in fact. A library of 310 papers selected from the American Mathematical Monthly, Mathematics Magazine and College Mathematics Journal.

40 years later, there is still no end to conversations and committee meetings, etc., but there appears to be a comprehensive effort under way to modernize instruction of undergraduate mathematics. In the November issue of the FOCUS newsletter, MAA has announced a two-year $900,000 grant from the NSF to construct the Mathematical Sciences Digital Library (MATHDL). MATHDL will comprise three related web-based components. The first part will be a new MAA publication - The Journal of Online Mathematics and its Applications (JOMA). JOMA is an official abbreviation. The journal will be available at The initial focus is, of course, on Calculus.

In August, 6 faculty went through the 600 calculus mathlets -- small learning units -- found and tested by students and from them selected 18 for JOMA and another 50 for LOMA, where "L" stands for "Library." LOMA -- the second component of MATHDL -- will contain a select collection of online learning materials, including mathlets, that have been classroom tested and peer reviewed. The third part will review and list commercially available learning materials. Gene Klotz - a founder and director of Math Forum and a co-PI for the current grant - has mentioned a suggestion to dub the latter COMA to complete an easily memorizable triple of acronyms.

Articles for the A Century of Calculus have been selected through a similar process. For the first volume, the editorial committee read 750 papers. Abstracts were prepared, and each paper received a vote Yes, No or Undecided. Each paper was read by at least two committee members. Those papers that received at least one vote of Yes or Undecided were given a third, and in some cases a fourth and fifth reading until some agreement was reached whether to include the paper in the collection or not. For the second volume the editorial committee reviewed more than 500 articles.

The first volume was published in 1969 and then reprinted along with the second volume in 1992. As expected of items in a book set, the two 1992 volumes look exactly the same on the outside: the same cover, the same design, the same paper, they even have about the same number of pages. The two volumes also have very similar chapter structures. In both, the quality of articles and their relevance are excellent. However they read differently. If I were an old grouchy man, I would probably grumble about how, compared to the good old days, things have changed to the worse. On the inside, the two volumes differ remarkably. Using the modern terminology, I would say that the second volume suffers from a relatively poor user interface. And here is perhaps a lesson to learn for the JOMA project.

As a matter of fact, I've been skimming A Century of Calculus looking for some good ideas for JOMA. Munroe's quote in the opening paragraph came from the first volume (p. 70). The first volume was easy to browse and easy to relate to without making long stops at any one article. Following are a few items that caught my attention and thoughts that had registered in my mind.

On page 48, R. G. Helsel and T. Radó pondered (MAA 55 1948, pp. 28-29) the question Can We Teach Good Mathematics To Undergraduates? They cited three ingredients of good mathematics: relevancy (calculus, for example, is overflowing with relevancy), rigor (all of the reasons must be given), and elegancy (not to deprive the students of the very thing that affords us our greatest pleasure).


Consistency, I thought, they did not mention consistency. Clearly the question at hand is that of good mathematics presentation. For a good course, consistency in style is a prerequisite. It could be a trifle. For example, if during the first few weeks Professor conditions the students by emphasizing important ideas with a chalk of red color, and then comes to a lesson without his implements, many students will be bound to miss something important.

Page 111. Relating to Finding Derivatives of Trigonometrical Functions by T. H. Hilderbrandt (AMM 25 (1918), pp. 125-126). Either sin(x + Δx) is expanded by the formula for the sine of the sum of two angles and the formula for 1 - cos(x) in terms of half angles is used, or the formula for the difference of two sines is used. Both of these latter formulæ have long since escaped the memory of the average sophomore student -- if they ever had lodging there -- and he practically accepts this part of the derivation on faith.


Can it be determined what an average sophomore remembers for sure? But interestingly enough the derivative of tan(x) needs less trigonometry than those of sin(x) and cos(x).

Page 114. Trigonometry From Differential Equations by D. E. Richmond (AMM 61 (1954), pp. 337-340). This note shows how analytic trigonometry may be developed in an elementary manner (with no use of infinite series) from the differential equation: d2y/dt2 + y = 0 ...


Well, talk about elementary differential equations. But all he uses is a simple harmonic motion. Set x = dy/dt. Then d2y/dt2 + y = 0 becomes dx/dt = -y.

Page 124. A Classroom Proof of lim sin(t)/t = 1 by S. Hoffman (AMM 67 (1960), pp. 671-672). The latter part of this string of inequalities (t < tan(t)) is usually determined by comparing the area of a unit-circular sector of angle t with the area of a triangle. If the finding of the area of a circular sector exhausts the students' reservoir of thinking, then an understanding of the remainder of the proof is lost. The following method of proof avoids the use of area.


It's funny, but how much can be done with a student who does not understand how to compute the area of a circular sector?

Page 126. Derivative of sin(θ) and cos(θ) by C. S. Ogilvy (AMM 67 (1960), p. 673). The derivatives of the sine and cosine functions may be presented with the aid of vectors.


Beautiful. Velocity is perpendicular to the line of motion, etc. Just a five-sentence article. Is it not a proof without words?

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

Page 153. A Peculiar Function by J. P. Ballantine (AMM 37 (1930), p. 250). A pie reposes on a plate of radius R. A piece of central angle θ is cut and put on a separate plate of radius r. How large must r be? Obviously r is a function of θ. It turns out to have the following formula:

r = 0 θ = 0
r = ½R·sec(½θ) 0 < θ≤ 90°
r = R·sin(½θ) 90°≤ θ ≤ 180°
r = R 180°≤ θ ≤ 360°

The function has one discontinuity at θ = 0, and its second derivative has various discontinuities.


Oops. Is not something wrong here? There are sure discontinuities at 90° and 180°. Something is wrong also with the inequalities. But that's OK. I know what he meant. A function that arises naturally but is not given by a single formula and not continuous. I had a simpler example. ... Oh, no. Dr. Bogomolny, you must learn your trigonometry. The inequalities are correct and the function is indeed continuous, except at r = 0. My example is still simpler.

(Remark. My thanks go to Ed Fisher who pointed out an unnoticed misprint in the original article inherited by the first version of the current page. It appears that my trigonometry does need a refresher. In the book, the third part of the definition of the function is cos(½θ). It's easy to check that the correct function is sine, as above.)

Page 251. Geometric Interpretations of Polynomial Approximations of the Cosine Function by E. R. Heineman (AMM 73 (1966), pp. 648-649). The polynomial approximations obtained from the power series expansion

cos(x) = 1 - x2/2! + x4/4! - x6/6! + ...

converge to the cosine function... In one sense, it is clear that the successive approximations become "better." The student, however, must be warned against the pseudo-truth: "Every time you add another term, the approximation gets better." The horizontal line y = 1 is a shabby approximation for the cosine curve, but it is never in error by more than 2, which is more than can be said about the later approximations.


Obviously away from the origin higher degree approximations overtake "slowly changing" polynomials of lower degrees. What's interesting is how many approximations can be obtained with the regular Java numeric facilities. 11 is the answer as I found out later.

(About the applet: Minimum and maximum values on the axes that define the view frame are clickable and also respond to the cursor being dragged in their vicinity. To increase a number, click or drag the cursor a little to the right of the central line of the number. To decrease it, click or drag to the left from the central line. The same holds for N - the number of displayed approximations.)

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

Page 287. On The Harmonic Mean of Two Numbers by G. Polya (MAA 57 (1950), pp. 26-28). x is in the interval [a, b], but otherwise unknown. Seek an approximate value p that minimizes the extreme error or the extreme relative error. Two problems:

Problem I. Find

and the value of p for which it is attained.

Problem II. Find

and the value of p for which it is attained.

For Problem I, p = (a + b)/2. For Problem II, the solution is a little less obvious and may be new, but is certainly very little known: p = 2ab/(a + b).


Well, certainly it is still very little known. Some time ago, for a project at Math Forum several mathematicians were looking for situations that give rise to the harmonic mean. No one thought of this one.

This was my take on browsing the first volume of A Century of Mathematics. I also enjoyed just handling the book. Not so with the second volume. First, all articles in the first volume came with a footnote reference to their source, year of publication in particular. It picked my interest to check the year just to confirm the obvious: the search for better ways is naturally an ongoing process. I expected the same quiet entertainment with the second volume. To my disappointment, articles did not carry the footnote which I had come to expect. (To be fair, I later discovered that all sources have been listed at the end of the book. But having to peek there all the time was a great annoyance.)

Second, in the second volume fonts changed from article to article. No, not very often, but often enough to cause a distraction. Third, in the second volume, articles differ in style. Some are preceded by the author's picture and an introductory paragraph, some carry the author's affiliation seal, some come unadorned. I am aware of the reason why that is so - the articles were photo reproduced from different journals - but this did not help. Casual leafing through the volume was rather tiresome. And this is to be regreted as the selection is quite excellent.

In Testing Understanding and Understanding Testing (CMJ 16, pp. 178-185) J. Pederson and P. Ross offer a sample of problems that require for their solution understanding of calculus concepts more than formula memorization or mechanical symbol manipulation. The following applet helps observe the relations between a function and its derivative and integral with no single formula involved.

(About the applet: Points on the graph below are draggable up and down. So that you can modify the function any other way you want. Simultaneously with the function, you may display its derivative and the integral as a function of its upper limit. If you drag the cursor away from the graph while the "Show tangent" box is checked, a short line will be displayed tangent to either graph of the function or the graph of the integral.)


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

Back to the MATHDL. To be useful, any software -- a stand-alone package or a library -- must, like a book, be coherent, reasonably comprehensive and consistent in its user interface. In this respect, learning materials in COMA will have a great advantage over freely available pieces from LOMA and even JOMA. The reviewing effort must be supplemented with purposeful development for the sake of consistency.

Fortunately, the JOMA proposal was written with proactive attitude in mind. Each topical area, like Calculus, will have a Table of Contents, and solicitations for missing pieces will be actively promoted through JOMA, now an online publication of the MAA. It will be hosted by the Math Forum. With Forum's experience at community building and publicity that comes from association with the MAA, the project has a very good chance for success.

Most certainly the many conversations, committee meetings, etc., devoted to the modernization of the undergraduate calculus course will not end with the planned inauguration of JOMA at the Joint Meeting in January. It would be to everyone's benefit were JOMA to become the focus of many of those conversations and meetings.


  1. A Century of Calculus, v. 1, T. M. Apostol et al, MAA, 1992
  2. A Century of Calculus, v. 2, T. M. Apostol et al, MAA, 1992

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