# Historical tidbits to liven up calculus classes

What are some examples of math history that can be mentioned in calculus classes, either to liven things up or to provide additional perspective / insight on the material being learned?

For example, when discussing complex numbers, one might tell the story about how Hamilton discovered the quaternions.

The great breakthrough in quaternions finally came on Monday 16 October 1843 in Dublin, when Hamilton was on his way to the Royal Irish Academy where he was going to preside at a council meeting. As he walked along the towpath of the Royal Canal with his wife, the concepts behind quaternions were taking shape in his mind. When the answer dawned on him, Hamilton could not resist the urge to carve the formula for the quaternions, $$i^2 = j^2 = k^2 = ijk = -1,$$

into the stone of Brougham Bridge as he paused on it.

What are your favorite math history stories to tell when teaching calculus? Are there any historical math problems, such as the Brachistochrone problem, that could be incorporated into a calculus class?

• Part of the reason that modern calculus texts have bloated up to 1000 pages is that they include lots of little anecdotes like this. So: collect a few of these textbooks, and look for the relevant anecdotes in all of them just before you go to your lecture. – Gerald Edgar Feb 15 '16 at 17:17
• Quaternions are a reasonable digression for the vector calculus, or third semester (USA) calculus. The relation to the dot and cross product as well as the story of how vectors replaced quaternions is interesting and relevant since we still use $\hat{i},\hat{j},\hat{k}$ despite the fact there are better notations (for example, $\hat{x},\hat{y}, \hat{z}$ is far more descripitve) – James S. Cook Jul 31 '16 at 2:19

What are some examples of math history that can be mentioned in calculus classes, either to liven things up or to provide additional perspective / insight on the material being learned?

You mention two different goals here. Personally, I have used anecdotes about and discussion of historical events/people in calculus courses to...

1. provide some context for course material within the broader history of human intellectual development.
2. provide some context for course material within the history and development of mathematics and/or physics and/or the sciences and/or problem-solving.
3. make a particular idea more memorable or enlightening by tying it to a specific story or discussion.
4. make class time more engaging /interesting/entertaining for the students.
5. show students that math is not a static body of knowledge, but rather an ever-changing social structure that happens to have a lot of "facts" in it.
6. satisfy myself personally, to be able to say, "I told them about $X$!"

It's rare that any particular anecdote/discussion will satisfy more than three of these goals; usually, I'm just satisfying a couple, depending on the audience. And occasionally, yes, on the spur of the moment, I might achieve only #4 or #6 by mentioning something that comes to mind that I hadn't planned on saying during class.

In the list that follows, I'll point out which particular goals (from my list) that the anecdote/discussion might achieve, based on my past experiences talking about these ideas in class. I'll also give a context for when the anecdote/discussion might be relevant in a calculus course.

• Leibniz/Newton and derivative notation. There has been discussion here about whether students should be required to use multiple notations. I find it worthwhile to point out the variety of notations for the derivative of a function, and describe where they come from, who wanted to use them, and why. Leibniz's philosophy made frequent use of monads, directly analogous to infinitesimals. His notation conveys his idea that a derivative is related to infinitesimal change in a function. Newton was developing calculus to be used in solving physical problems, with functions of time. His notation lends itself nicely to those problems. Lagrange's notation is more concise but not necessarily as clear. Euler's notation generalizes to other things but is a bit overblown for just Calc 1. In a vector calculus course, you could even engage students in a discussion about the "operator" notations of divergence and the gradient, and point to overarching concepts of functional analysis. In the context of an introductory calculus course, I think this hits goals #2,3,5.
• "Leibniz got the product rule wrong! Har har!" I cannot find the reference at the moment, but I distinctly remember a marginal note in the Stewart calc text. When it introduces the product rule, it has a short aside about how Leibniz first incorrectly guessed, "as you might, dear reader", that $(fg)'=f'g'$. I am not at all against the idea of pointing out how mathematicians have corrected themselves, individually and collectively, over time. But I remember feeling like this was an overt jab at Leibniz. Nowhere in the text does it mention anyone else's mistakes, nor does it say that Leibniz fixed it and deserves much credit for developing the calculus, nor does it describe how Newton and his cronies browbeat the others into accepting his as the primary/premiere publication. But anyway ... I now like to point out this example and show that, yes, humans get things wrong, but it's all about our progress in figuring things out, not just "getting the right answer" right away. In an intro calc course, I think this hits goals #1,2,3,5.
• Should "L'Hopital's Rule" be "Bernoulli's Rule"? We train students to recognize when and how to use L'Hopital's Rule. But very few of them know that L'Hopital was paying the Bernoullis for personal tutoring, and he assumed that anything they developed in their lessons was ripe for publishing under his own name. This can lead to an interesting discussion of intellectual property and the "body" of mathematical knowledge, as well as serve as a way to point out the historical influence of the Bernoulli family throughout mathematics. In an intro calc course, I think this can hit all of the listed goals.
• Euler and the Basel Problem. This is one of the historical touchstones in evaluating infinite sums exactly, and it's also the event that birthed Euler onto the mathematical landscape. Moreover, there are new proofs of this fact published all the time (a handful in the last year or two, at least) over 250 years later. A calc 2 course can follow some of the big ideas in Euler's proofs (he had three, actually), and, in the past, I have set aside 30 minutes of class time to outline one of the proofs. This story establishes Euler as the genius that he was, and shows that some results never really "get old". Personally, I've mostly used this for goal #6, but it also addresses #3,4.
• Nicole Oresme's proof (14th century) that the harmonic series diverges. This is, in fact, the "standard" proof, showing that blocks of terms add to $>1/2$, so the whole series is $>1/2+1/2+\dots$. But hardly anyone points out that this proof is due to Oresme in the mid-1300s and was lost for the next few centuries during the Dark Ages (when not much science or math was really going on, at least in the Western world) and was "rediscovered" by Pietro Mengoli around when he posed the Basel Problem. In a calc 2 course, this hits goals #1,3,5.
• "The reinvention of the trapezoid rule" in 1994. This incident brought about some vitriol between the paper's author and those who wrote letters to the associated journal and, as I've noticed, has inspired some not-so-nice discussion about how most scientists "don't know how to do math" or are "uninterested" in spending the time to learn the math of their discipline (I mean, just look at the comments on that reddit thread I linked to). I'd recommend avoiding those discussions. Instead, I like to bring up this example when we study numerical approximation of integrals and say that some ideas are just so important and fundamental that humans will keep "rediscovering" them, apparently. (And yes, you can follow this up to say, "It behooves you, future scientists, to learn whatever you can so that you don't spend time reinventing the wheel," if you'd like.) In a calc 1/2 class, this hits goals #1,2,3.
• Weierstrass' Function and "the lamentable scourge of continuous functions with no derivatives." This story is more worthwhile in a class with math-oriented students, potential future majors who will take Real Analysis. (I'd avoid anything more than a precursory mention with a class of non-math majors.) But when you bring up the idea of "differentiable implies continuous" but not the converse, you can point to Hermite's quote from 1893 -- ""I turn with terror and horror from this lamentable scourge of continuous functions with no derivatives." -- as well as to Weierestrass' pathological function. You could even use this to preview some notions about infinite cardinalities and say how "almost every" $f:\mathbb{R}\to\mathbb{R}$ is not even continuous. In an honors calc class, this can hit goals #2,3,5.
• Re Leibniz getting the Leibniz rule wrong: mathoverflow.net/questions/181422/… – Ben Crowell Sep 21 '14 at 2:55
• The 1300s were well after the end of the Dark Ages. Oxford, Paris, Cambridge, and Padua had universities already. Merchants could safely travel the length and breadth of England. – Jasper Oct 18 '14 at 5:18
• I wonder if someone has made an exhaustive study of basic numerical integration being rediscovered in various quasimathematical sciences. I bet this sort of oversight is perpetual and systemic. – James S. Cook Jul 31 '16 at 2:16
• Regarding the Leibniz product rule story, here is what Stewart (2008, p. 183) claims (without any evidence): "By analogy with the Sum and Difference Rules, one might be tempted to guess, as Leibniz did three centuries ago, that the derivative of a product is the product of the derivatives. We can see, however, that this guess is wrong by looking at a particular example. ... The correct formula was discovered by Leibniz (soon after his false start) and is called the Product Rule." – Kenny LJ Oct 8 '18 at 4:21

There are SO many. See e.g. Fred Rickey's website. My two favorites are

• The story of Hobbes' response to Torricelli's computation of the volume of the so-called Gabriel's Horn (see e.g. Calculus Gems for this), and
• Leibniz' wrong product rule, which I had a handout from Rickey on, but which I can't find now - perhaps it's linked here somewhere.

There is a fabulous story that is bound to captivate the students about Cauchy's alleged "error" in claiming that a convergent sequence of continuous functions necessarily converges to a continuous function. This story is recycled in many history and calculus textbooks. Detlef Laugwitz was one of the first to show that this anecdote about Cauchy is anachronistic and involves imposing modern notions on 19th century work. Laugwitz points out that Cauchy clarified the correct hypothesis in a 1853 paper. A recent study of the issue on Cauchy's own terms can be found here.

Another great tale is about the Greek philosopher Hippasus who is usually credited with the discovery that $\sqrt{2}$ is irrational. For this, according to legend, he was thrown overboard by enraged Pythagorians. Many students today are cautious to avoid a similar fate at the hand of enraged Weierstrassians, as argued in this article.

I think a more intriguing idea is to understand why more popular textbooks like the canonical Stewart textbook present materials paying no attention to historical development. In doing so, students are exposed to mathematics as a completely abstract discipline that exists outside of human beings. While some work has been done to correct this overtly White European reconstruction of history, much more needs to be done.

Historical tidbits is perhaps not the only way to accomplish this, but instead maybe we can communicate the messy development of the discipline that has taken place over thousands of years all over the world. Toeplitzs' genetic approach is a starting point, but so are Hairer and Wanner's more recent text on analysis, Edwards text The Historical Development of the Calculus, and in particular David Perkins Calculus and Its Origins from the MAA is an inspirational textbook that I continue to use with my undergraduate Calculus classes.

With many mathematics educators discussing issues of race, gender, and equity, it is important to understand that people other than white europeans have accomplished enormous feats relevant to the calculus curriculum. Two examples might be ibn Al Haytham and his computation of solids of revolution and Jyesthadeva and his approximation of pi through the use of an alternating series. From a more philosophical standpoint, see Serres The Birth of Physics, though this is pretty white as well.

More than anything, we should continue to see the curriculum as a work in progress rather than a static set truth as presented in its best image through the work of those like Stewart and Larson whose texts cost hundreds of dollars and are inaccessible to many. Reorganize these ideas from what you believe is a more impassioned presentation of mathematics and share with all!

The controversy between English and Continental mathematicians about who should get credit for inventing Calculus, as described (in a journalistic sort of way) in the book "The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time" by Jason Socrates Bardi. This brings out the human element in mathematics.

Some applications of calculus of types not covered in most textbooks might be interesting. For example, few students growing up in the digital age realize just how sophisticated analog computers had become by the end of World War II. The German rocket scientists used an analogue integrator of acceleration to obtain speed called an integrating accelerometer to control engine shut-off for their V2 rockets.

For Calc III, the planimeter (an analogue device for calculating area by tracing around a perimeter) can be introduced in the context of discussing Green's Theorem (an idea that I got from the third edition of Stephen Krantz's wonderful book "How to Teach Mathematics").

• Planimeters are cool. – kcrisman Oct 31 at 2:07

I love the answers already here, but just wanted to add: I have always rather enjoyed the "May we not call them the Ghosts of departed Quantities?" quote of George Berkeley re: infinitesimals. I found it fascinating that infinitesimals/limits had been pretty controversial at one point; since many students also struggle with these notions this has seemed to hearten them as well.

See more here: http://en.wikipedia.org/wiki/The_Analyst

The Maya had a place-value number system with a "numeral" to represent zero long before those in the other hemisphere.

There is evidence that "Pascal's Triangle" was known and used in China long before Pascal lived.

It is likely the ancient Egyptians knew an equivalent to the "Pythagorean Theorem."

Why do we refer many of these important results only be the name associated with a European who discovered or invented or was the first to publish them?

• Pascal's triangle was associates with him because he used it often, not because it was claimed he invented it. – vonbrand Feb 14 '16 at 18:08

The story that DeCarte came up with the idea of rectangular coordinates by wondering if he could describe the position and path of a bug on the ceiling.

The story that Gauss was acting up at school, and his teacher tried to punish him by asking him to find the sum of the integers from 1 to 100 thinking it would keep him busy for a LONG time. It only took him a few minutes, and the only thing he wrote on his slate was the result.

Then we can talk about which of these types of stories are fact, fiction, or myth and who gets to write the history.

Oh, one more. Sonia Kovalevsky learned calculus at a young age because her father wall-papered he bedroom with the pages of a calculus text.

• Like these. These are lighthearted and fun. – guest Mar 12 '18 at 14:33

Complex numbers and quaternions are very different beasts, don't go this way. Perhaps explain how complex numbers came to be recognized by solution to cubics.

An interesting historical tidbit is Cantor's attitude to infinitesimals, which he variously described as "cholera bacillus of mathematics", "paper numbers", and even "abomination" (see P. Ehrlich's article from 2006). Cantor claimed that he proved infinitesimals to be inconsistent. He even published a paper to the effect. The reportedly great philosopher Russell was convinced by Cantor's claim and even reported in his book Principles of Mathematics that infinitesimals are inconsistent. Maybe they were right...

One of my favorites is this quote of Abel, identifying the notion of analytic function. (emphasis mine)

My work in the future must be devoted entirely to pure mathematics in its abstract meaning. I shall apply all my strength to bring more light into the tremendous obscurity which one unquestionably finds in analysis. It lacks so completely all plan and system that it is peculiar that so many have studied it. The worst of it is, it has never been treated stringently. There are very few theorems in advanced analysis which have been demonstrated in a logically tenable manner. Everywhere one finds this miserable way of concluding from the special to the general, and it is extremely peculiar that such a procedure has led to do few of the so-called paradoxes. It is really interesting to seek the cause.

In analysis, one is largely occupied by functions which can be expressed as powers. As soon as other powers enter—this, however, is not often the case—then it does not work any more and a number of connected, incorrect theorems arise from false conclusions. I have examined several of them, and been so fortunate as to make this clear. ...I have had to be extremely cautious, for the presumed theorems without strict proof... had taken such a stronghold in me, that I was continually in danger of using them without detailed verification.

https://en.wikiquote.org/wiki/Niels_Henrik_Abel

Some of the previous answers are of the nice-to-us-teachers-but-irrelevant-to-students type. I favor more concrete tidbits:

In calculus it makes sense to use exponent notation for radicals and reciprocals, because derivative rules are much more straightforward. Indeed, $(x^{1/2})'$ and $(x^{-1})'$ are concrete cases of the power rule. Compare this to learning yet one more formula for $\sqrt{x}'$. Students readily assimilate this "trick" when I explain that the radical sign is (truly is!) a stylized version of the letter r.

• "Some of the previous answers are of the nice-to-us-teachers-but-irrelevant-to-students type. I favor more concrete tidbits." Yes. Common pedagogical issue and not just on this question. – guest Mar 12 '18 at 14:21

I think the stories here are about rather hard or minute issues that would distract the class. Quaternions? Really? For kids learning sqrt(-1)? I would actually avoid these stories unless you have kids that are just crushing it on the topic and have time to waste.

The one story that seemed simple and friendly was the bug on the ceiling for Descartes. A rectangular ceiling has a very Cartesian look to it. Things like that are good. Minutia of Leibniz versus Newton, or L'Hopital is wrong name, no. That's confusion.

What is useful instead is to emphasize how the courses have had value in the life of the instructor. And you don't need to do detailed applied problems. Just let them know that knowing trig and calc have been personally helpful to you in physics classes and in the military and oil exploration (or whatever from your personal life). Just that knowing the stuff helped you earn some bread.