From American Mathematical Monthly, March 1988
Calculus with Analytic Geometry. By George F. Simmons. McGraw-Hill Book
Company. New York, 1985. xiii + 950 pp. $46.95.
Department of Mathematics and Computer Science, DePauw University,
Greencastle, Indiana 46135
This is about calculus books, and there are seven important conclusions.
Let us go back to the beginning and look at the first calculus book, Analyse des
Infiniment Petits, pour L'intelligence des Lignes Courbes, published in Paris in
-1696 and written by Guillaume François Antoine L'Hospital, Marquis de Sainte-Mesme,
Comte d'Entrement, Seigneur d'Orques, etc. To almost everyone, L'Hospital is just a
name attached to L'Hospital's Rule and almost no one knows anything about him. His
memory deserves better. He displayed mathematical talent early, solving a problem
about cycloids at fifteen, and he was a lifelong lover and supporter of mathematics
who, unfortunately, died young, at the age of 43. Also, L'Hospital was no. mean
mathematician. He published several papers in the journals of the day, solving
various nontrivial problems. I know that I could not have found, as he did, the
shape of a curve such that a body sliding down it exerts a normal force on it always
equal to the weight of the body. Further, as Abraham Robinson has written,
According to the testimony of his contemporaries, L'Hospital possessed a very
attractive personality, being, among other things, modest and generous, two
qualities which were not widespread among the mathematicians of his time.
His book was a huge success. There was a second edition in 1715, and there were
commentaries written on it. I have the 1781 edition, with additions made by another
author. Not many textbooks last almost 100 years. Birkhoff and MacLane is not yet
50 years old. L'Hospital's book is about differentials and their applications to
curves and the style is exclusively geometrical. There are not many equations, but
there are an awful lot of letters and pictures, just as I remember in my tenth grade
geometry text. Mathematics was geometry then, and mathematicians were geometers.
There are many things not in the book. There are no sines or cosines, no
exponentials or logarithms, only algebraic functions and algebraic curves. There
are also no derivatives, only differentials. Here is L'Hospital's proof of
L'Hospital's Rule, using differentials. In Figure 1, points on the graph of
f(x)/g(x) are found by dividing lengths of abscissas, except at a. But if you dx
past a, the point on the graph will be dfldg. Since dx is infinitesimal, that ratio
gives you the point at a. Is that not nice?
It took some time for calculus to become generally taught in colleges. Eventually
it made it, and calculus textbooks began to appear in the nineteenth century. I
have a copy of one, Elements of the Differential and Integral Calculus, by Elias
Loomis, LI. D., Professor of Natural Philosophy at Yale College. His calculus was
first published in 1851, and my copy of it is the 1878 edition. It sold in excess
of 25,000 copies, so it must reflect accurately the style and content of calculus
teaching of the time. Just as with L'Hospital, the differential was the important
idea. Loomis derived the formula for the differential of x" with no use of the
binomial expansion, (d(xy)/xy = dx/x + dy/y, d(xn)/xn = dxlx + + dxlx, add and
simplify) and his proof of L'Hospital's Rule was short, simple, and clear, and also
one which does not appear in modern texts because it fails for certain pathological
examples. Also, Loomis put all of his formulas in words, italicized words. After
deriving the formula for the differential of a power of a function, he wrote
The differential of a function affected with any exponent whatever is the continued
product of the exponent, the function itself with its exponent diminished by unity,
and the differential of the function.
It was a good idea. It is also probably a good idea to do as L'Hospital and Loomis
did and talk about differentials instead of derivatives whenever possible. Little
bits of things are easier to understand than rates of change. It is a still better
idea to strive for clarity and let students see what is really going on, which is
what Loomis did, rather than putting rigor first. But nowadays authors cannot do
that. They must protect themselves against some colleague snootily writing to the
publisher, "Evidently Professor Blank is unaware that his so-called proof of
L'Hospital's Rule is faulty, as the following well-known example shows. I could not
possibly adopt a text with such a serious error." It is a shame, and probably
inevitable that calculus books are written for calculus teachers, but I have
CONCLUSION #1: CALCULUS BOOKS SHOULD BE WRITTEN FOR STUDENTS.
It would be worth a try. Calculus Made Easy by Silvanus P. Thompson was quite
successful in its time, which ran for quite a while. The second edition appeared in
1914, and my copy was printed in 1935. It is still in print. The book has a
What one fool can do, another can.
and a prologue:
Considering how many fools can calculate, it is surprising that it should be thought
either a difficult or a tedious task for any other fool to learn how to master the
same tricks. ...
Being myself a remarkably stupid fellow, I have bad to unteach myself the
difficulties, and now beg to present to my fellow fools the parts that are not hard.
Master these thoroughly, and the rest will follow. What one fool can do, another
Chapter 1, whose title is "To Deliver You From The Preliminary Terrors" forthrightly
says that dx means "a little bit of x." Thompson did not include L'Hospital's Rule.
Both Loomis and Thompson are like L'Hospital when it comes to giving applications
and examples: they are all almost entirely geometrical. Loomis's applications of
maxima and minima are all about inscribing and circumscribing things, and so are all
of Thompson's except one. In fact, all three books are full of geometry. Thompson
concluded with arc length and curvature, Loomis had involutes and evolutes, cusps
and multiple points, and lots of curve sketching. Did you know that the asymptote
to y3 = X3 + X2 is y = x + 1/3? 1 didn't until I read Loomis. It is nice to know.
There must have been some reason why calculus books for more than 200 years taught
so much geometry. Mathematics may no longer be synonymous with geometry, but we
have discarded, wrongly I think, the wisdom of the ages, and I have concluded
CONCLUSION #2: CALCULUS BOOKS NEED MORE GEOMETRY.
Before writing this essay, I examined 85 separate and distinct calculus books. I
looked at all of their prefaces, all of their applications of maxima and minima, and
all of their treatments of L'Hospital's Rule. By the way, I found five different
spellings of L'Hospital. There were the two you would expect, and Lhospital, as
L'Hospital sometimes spelled his name. In addition, one author, not wanting to take
chances, had it L'Hôspital, and one thought it was Le Hospital. Why are there so
many calculus books, and why do they keep appearing? One could be cynical and say
that the authors are all motivated by greed. But I do not think so. I think that
authors write new calculus books because they have observed that students do not
learn much from the old calculus books. Therefore, prospective authors think, "if I
write a text and do things properly, students will be able to learn." They are
wrong, all of them. The reason for that is
CONCLUSION #3: CALCULUS IS HARD.
Too hard, I think, to teach to college freshmen in the United States in the 1980s,
but that is another topic.
If you plot the books' numbers of pages against their year of publication, you have
a chart in which an ominous increasing trend is clear. The 1000-page barrier, first
pierced in 1960, has been broken more and more often as time goes on. New highs on
the calculus-page index are made almost yearly. Where will it all end? We can get
an indication. The magic of modern statistics packages produces the least-squares
line: Pages = 2.94 (Year) - 5180, showing that in the middle of the next millennium,
the average calculus book will have 2,270,pages and the longest one, just published,
will have 3,783 pages exclusive of index.
Why do we need 1000 pages to do what L'Hospital did in 234, Loomis in 309, Thompson
in 301, and the text I learned calculus from, used exclusively for four whole
semesters, 14 semester-hours in all, in 416? There are several reasons. One, of
course, is the large number of reviewers of prospective texts. No more can an
editor make up his mind about the merits of a text, it has to go out to fifteen
different people for opinions. And if one of them writes that the author has left
out the tan(x/2) substitution in the section on techniques of integration, how can
he or she do that, we won't be able to integrate 3/(4 + 5 sin 6x), how can anyone
claim to know calculus who can't do that; isn't the easiest response to include the
tan(x/2) substitution? Of course it is, in it goes, and in goes everything else
that is in every other 1000-page text. It is impossible to escape
CONCLUSION #4: CALCULUS BOOKS ARE TOO LONG.
Another reason for the length is the current mania for Applications. If you go to
Books in Print and look in the subject index under "Calculus" what you see is
The Usefulness of Calculus for the Behavioral, Life, and Managerial Sciences
Essentials of Calculus for Business, Economics, Life Sciences, and Social Sciences
and many, many similar titles. Now authors have to explain, with examples, what
marginal revenue is, and consumer surplus, and what tracheae are whereas in the old
days, all their readers knew what a cone was. A third reason is the supposed need
to be rigorous. Now we see statements of L'Hospital's Rule that take up half a page
and proofs of it that go on for three pages. My 416-page calculus book never even
mentioned L'Hospital's Rule, and I never felt the lack. Its author never proved
that the derivative of x' was ex'-, but I was willing to believe it. Trying to
include everything and trying to prove everything makes for long books. Everything
gets longer. Prefaces used to be short, a page or less. Now they are five and six
pages, hard sells for the incredible virtues of the text that follows, full of
thanks to reviewers, to five or six editors, to wives, to students, even to cats.
Let me return to "applications." There aren't many, you know. In the 85 calculus
books I examined, almost all of them had the Norman window problem-the rectangle
surmounted by a semicircle, fixed perimeter, maximize the area. The semicircle
always "surmounts." This is the sole surviving use of "surmounted" in the English
language, except for the silo, a cylinder surmounted by a hemisphere. Only one
author had the courage to say that the window was a semicircle on top of a
rectangle. All the books had the box made by cutting the corners out of a flat
sheet, all have the ladder sliding down the wall, all had the conical tank with
changing height of water, all had the tin can with fixed surface area and maximum
volume, all had the V-shaped trough, all had the field to fence, with or without a
river flowing (in a dead straight line) along one side, all had the wire-usually
wire, but sometimes string-cut into two pieces to be formed into a circle and a
square, though some daring authors made circles and equilateral triangles. There
are only finitely many calculus problems, and their number is very finite.
"Applications" are so phony. Ladders do not slide down walls with the base moving
away from the wall at a constant rate. Authors know the applications are phony.
One book has the base of the ladder sliding away from the wall at a rate of 2 feet
per minute. At that rate, you could finish up your painting with time to spare and
easily step off the ladder when it was a foot from the ground. Another author has
the old run-and-swim problem-you know, minimize the time to oet somewhere on the
other side of the river-with the person able to run 25 feet per second and swim 20
feet per second. That's not bad for running (it's a 3:31.2 mile), but it is super
swimming, 100 yards in 15 seconds, a new world's record by far. There are no
conical reservoirs outside of calculus books. Real reservoirs are cylindrical, or
perhaps rectangular. The reason for this is found in the texts: in the problems,
the conical reservoirs usually have a leak at the bottom. Tin cans are not made to
minimize surface area. I could give any number of examples of absurd applications
in which businessmen "observe" the price of their product decreasing at the rate of
$1 per month, or where the S. D. S. (remember them?) "find" that staging x
demonstrations costs $250x'. Why will authors not be honest and say that these
artificial problems provide valuable practice in translating from En-lish into
mathematics and that is all they are for? Surely they cannot disagree with
CONCLUSION #5: FIRST-SEMESTER CALCULUS HAS NO APPLICATIONS.
Before getting to my next conclusion, here is my favorite "application."
A cow has 90 feet of fence to make a rectangular pasture. She has the use of a
cliff for one side. She decides to leave a 10 foot gap in the fence in case the
grass should get greener on the other side. Find .. . .
Hardly any authors dare to do that. Calculus books are Serious. The text from
which that problem came was titled Calculus Without Analytic Geometry and it is no
surprise that it did not catch on.
The existence of all those calculus books with "Applications" in their titles
implies a market for them, There must be students out there who are being forced to
undergo a semester of calculus before they can complete their major in botany and
take over the family flower shop. I cannot believe that any more than a tiny
fraction of them will ever see a derivative again, or need one. Calculus is a
splendid screen for screening out dummies, but it also screens out perfectly
intelligent people who find it difficult to deal with quantities. I don't know
about you, but I long ago concluded
CONCLUSION #6: NOT EVERYONE NEEDS TO LEARN CALCULUS.
The book by Simmons is a fine one. It was written with care and intelligence. It
has good problems, and the historical material is almost a course in the history of
mathematics. It is nicely printed', well bound, and expensive. Future historians
of mathematics will look back on it and say, "Yes, that is an excellent example of a
late twentieth-century calculus book." This leads to my last conclusion
CONCLUSION #7: THAT'S ENOUGH ABOUT CALCULUS BOOKS.