# Mathematical Time Capsules: Historical Modules for the Mathematics Classroom

Dick Jardine
Amy Shell-Gellasch
Series: MAA Notes
Volume: 77
Edition: 1
Pages: 304
https://www.jstor.org/stable/10.4169/j.ctt5hh8hc

1. Front Matter
(pp. i-vi)
2. Preface
(pp. vii-x)
(pp. xi-xviii)
4. 1 The Sources of Algebra
(pp. 1-6)
Roger Cooke

Nowadays we recognize written algebra by the presence of letters (calledvariables) standing for unspecified numbers, and especially by the presence of equations involving those letters. These two features—letters and equations—reveal thetechniquesof algebra, but algebra itself isnotthese techniques. Rather, algebra consists of problems in which the goal is to find a number knowing certain indirect information about it. If you were told to multiply 7 by 3, then add 26 to the product, you would be doing arithmetic, that is, you would be given not only the data, but also told which operations you...

5. 2 How to Measure the Earth
(pp. 7-16)
Lawrence D’Antonio

Who first determined the size of the Earth? How did they do it? These fundamental questions arise in studying early Greek, Indian and Islamic mathematical astronomy. In this article we look at the attempts of Eratosthenes, Posidonius, and al-Bīrūnī to determine the circumference of the Earth and ways to use this topic in the classroom. These calculations use only basic knowledge of geometry and trigonometry, so that instructors in many different courses can include this topic in their syllabus. It would be appropriate to discuss the problem in a high school or college geometry class, in a precalculus class, a...

6. 3 Numerical solution of equations
(pp. 17-22)
Roger Cooke

Methods of solving polynomial equations lie at the heart of classical algebra. There are two interpretations of the problem of solving an equation, leading to two different approaches to its solution. In most courses, the emphasis is on the structure of the equation and finding a way to express the roots as aformulain terms of the coefficients. The simplest example of such a formula is the quadratic formula, which gives the solution of the equationax2+bx+c= 0 as$x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}.$

This approach is elegant and leads to some exceedingly profound mathematics. However, for one who actually needs to...

7. 4 Completing the Square through the Millennia
(pp. 23-28)
Dick Jardine

Solving quadratic equations is a topic relevant to modern mathematics instruction, as it has been for thousands of years. As we start the 21st century, more often than not students will use calculators and computer algebra systems to solve quadratics. Today, we associate solving quadratics with curves (parabolas) rather than rectangles and squares (even though the word quadratic is from the Latinquadratum, a four-sided figure). A centuries old method which hopefully will survive in classrooms in this millennium is the method of completing the square. Understanding the process of completing the square is important for our students, for a...

8. 5 Adapting the Medieval “Rule of Double False Position” to the Modern Classroom
(pp. 29-38)
Randy K. Schwartz

The rule of double false position is an arithmetical procedure for evaluating linearly-related quantities. The method does not rely on variables or equations, but is based instead on interpolating between, or extrapolating from, two guesses, or suppositions. Although the technique is seldom mentioned today in North American curricula, it was routinely used in much of Europe, Asia, and North Africa from medieval times to the 19th Century, and is still taught in many classrooms there today. Historically, the approach was especially convenient for practical tradesmen whose knowledge did not normally extend to a mastery of algebra; they could pull the...

9. 6 Complex Numbers, Cubic Equations, and Sixteenth-Century Italy
(pp. 39-44)
Daniel J. Curtin

The complex numbers are important in modern mathematics and science, yet they receive almost no attention in the modern curriculum, which is heavily weighted towards preparation for the Calculus. Most pre-calculus treatments of the complex numbers give no insight into where they came from. They are mainly seen as supplying a full set of roots for polynomials that do not have all real roots. In fact, they first arose because they were needed to find real roots for cubic equations, precisely in the case where all three roots are real. The material in this article can be used anywhere complex...

10. 7 Shearing with Euclid
(pp. 45-54)
Davida Fischman and Shawnee McMurran

The Pythagorean Theorem is one of those intriguing geometric concepts that provide a never-ending source of ideas at all levels. Proofs of this theorem abound in print¹, and one wonders whether humans will ever stop looking for yet another. Indeed, it would be unusual for a student who has taken algebra or geometry not to have been exposed to at least one proof of the theorem, but how many have had occasion to explore the proof appearing in Euclid’sElements? In this proof, Euclid introduces a clever and elegant application of the concept ofshearing. It is a proof that...

11. 8 The Mathematics of Measuring Time: Astronomical Timekeeping and the Sinking-Bowl Water-Clock in India
(pp. 55-62)
Kim Plofker

In today’s world of electronic clocks and universal calendars, it’s easy to forget how important mathematics used to be just for the fundamental task of figuring out what time it was. The standard rigorous approach to the problem involved applying trigonometry to observed positions of the sun or the stars, as described below (“In the Classroom”). But several simpler methods were also developed for use when observations were unavailable or calculation was unappealing. One such practical device was the sinking-bowl water-clock, used for many centuries in India. Students (and teachers) will be impressed by how easy such a clock is...

12. 9 Clear Sailing with Trigonometry: Navigating the Seas in 14th-Century Venice
(pp. 63-72)
Glen Van Brummelen

Does anyone care about trigonometry? Certainly many of our students don’t, aside from the exigency of getting through their exams. As mathematics teachers, we have passion for our subject for its own sake — but we often justify ourselves to our students in terms of what the mathematics can accomplish elsewhere. For trigonometry as for many other topics, this takes the form of the widespread “word problems”: how high is that pine tree across the street? How far did that motorboat travel when it went across the lake? And here we reach a crucial pedagogical problem: few of us really...

13. 10 Copernican Trigonometry
(pp. 73-88)
Victor J. Katz

In most trigonometry courses, the instructor begins by defining the sine, cosine, and tangent of an angle as ratios of certain sides in an appropriate right triangle. She then proceeds to calculate, using elementary geometry, the sine, cosine and tangent of angles of 30°, 45°, and 60°. But once students need to calculate the sine of 27°, they are told to punch some buttons on their calculators. What do students think happens when they do that? Do they imagine that somewhere inside the calculator, someone draws a miniature right triangle with one base angle 27°, then measures the sides and...

14. 11 Cusps: Horns and Beaks
(pp. 89-100)

This is the mathematical tale of a cusp in the shape of a bird’s beak. Although precalculus and calculus courses must stress the idea of function over that of equation, they nevertheless include a number of important topics concerning polynomial equations in two variables, including implicit differentiation and the study of conic sections. Whereas polynomial functions of one variable have very simple graphs, the graphs of polynomial equations inxandy— even those of relatively low degree — can exhibit wonderfully exotic features.

The story of the bird’s beak can be used to enrich a course in analytic...

15. 12 The Latitude of Forms, Area, and Velocity
(pp. 101-106)
Daniel J. Curtin

Long before the calculus arrived a medieval philosopher, Nicole Oresme, developed what he called thelatitude of forms, a graphical representation that sheds light on the fundamental connection between area and what we now call the integral. In a calculus course, the latitude of forms can be used to introduce the idea of the integral as area, while simultaneously introducing the idea that the distance traveled is the integral of velocity. Of course the two ideas can be addressed separately, if you prefer. In that case, the latitude of forms might be used to connect the two. In any event,...

16. 13 Descartes’ Approach to Tangents
(pp. 107-110)
Daniel J. Curtin

While the modern version of tangents is central to the ideas of the differential calculus, I find students can profit from seeing an earlier and different approach. This minor detour also has the amusing aspect of using quite modern technology to help with an old problem. I use this material at the beginning of Calculus 2, when the students are fairly comfortable with the modern definition of derivative. One class period is used to present Descartes’ approach, then students receive a take-home assignment.

InLa Géometrie(1637) [2] René Descartes presents his general method of drawing a straight line to...

17. 14 Integration à la Fermat
(pp. 111-116)
Amy Shell-Gellasch

Move over Riemann and make room for Fermat! Most textbooks on the integral calculus focus heavily on the Riemann integral when introducing integration. This method is very effective in transitioning students from the finite (or macro) world of finding area geometrically to the infinite (or micro) world of finding area by integration. Once the notation and abstract idea of an area made up of an infinite number of infinitely thin slices is mastered, most textbooks move directly on to integration techniques. Finding areas using rectangles is usually not mentioned again except in review, to help students visualize a more difficult...

18. 15 Sharing the Fun: Student Presentations
(pp. 117-122)
Amy Shell-Gellasch and Dick Jardine

Advocates of incorporating the history of mathematics in teaching mathematics do so believing that providing a human element may spark student interest in mathematics. Incorporating biographical sketches or historical anecdotes into instruction has the potential to enhance student interest, with the hope that interested students will learn more readily and retain the content longer. The learning value of the historical activities can be enhanced when explored and presentedby the studentrather than presentedto the studentby the instructor or textbook. An effective way to have your students deepen their knowledge of mathematics through its history is to have...

19. 16 Digging up History on the Internet: Discovery Worksheets
(pp. 123-126)
Betty Mayfield

So you want to include some history of mathematics in your upper-level courses, but you just can’t imagine how you can possibly fit anything else in this semester. How will you get to all the topics you want to cover, and still have time for some history?

Instead of giving a lecture on a history topic, or on the name behind a famous theorem, why not let students find the information themselves? Using a discovery worksheet is fun, saves class time, and encourages students to learn things on their own. Some of the answers may be in their own textbooks,...

20. 17 Newton vs. Leibniz in One Hour!
(pp. 127-132)
Betty Mayfield

In our college, we teach a quick, one-credit Calculus Workshop course for students who have received credit for taking first-semester calculus elsewhere (in high school or at another college) but who need a brief introduction to some specific topics they may not have seen before. And so we spend one class on using a computer algebra system, one class on Euler’s Method, one class on writing about mathematics … a whirlwind tour of a variety of topics our regular Calculus I students see in more depth. The class meets for one hour and fifteen minutes twice a week during the...

21. 18 Connections between Newton, Leibniz, and Calculus I
(pp. 133-138)
Andrew B. Perry

Calculus, like most other well-established branches of mathematics, did not originally appear in the same form as it occurs in modern textbooks. Many mathematicians contributed to the development of calculus over many centuries, using widely varying notation and languages. A proper history of the subject can easily consume a book [1].

Although a thorough study of the history of calculus is completely unnecessary for an introductory calculus student, it is nevertheless of some interest for such students to see an overview of this subject’s fascinating and colorful history. Today’s calculus students will no doubt consider original papers somewhat cryptic at...

22. 19 A Different Sort of Calculus Debate
(pp. 139-150)
Vicky Williams Klima

As in most subjects, the historical significance credited to certain events in the development of calculus depends significantly on the historian giving the account. While thinking about how I should interpret selected historical events when presenting them to my first semester calculus classes, I realized that such a decision was unnecessary; my students could determine the appropriate interpretation for themselves through in-class debates. The debate project focuses on two topics: Fermat’s method of maxima and minima and Barrow’s theorem.

Debates allow students to actively participate in the learning process. David Royse [9] proposes that student learning is at its best...

23. 20 A ‘Symbolic’ History of the Derivative
(pp. 151-158)
Clemency Montelle

How often do you have to tell your students to brush up on their notation? When they have dropped limit notation, forgotten critical modulus signs, mixed up their integrals, muddled up their derivatives, how do you convey to them the importance of recording it right?

What of your exasperation as they fail to appreciate the precision that mathematical notation affords them—notation which has been developed and refined over centuries, and notation that will continue to be improved for centuries more. Indeed, mathematics is the one subject in which they can really expressexactlywhat they mean. How can we...

24. 21 Leibniz’s Calculus (Real Retro Calc.)
(pp. 159-168)
Robert Rogers

To many students, differential calculus seems like a set of rules to be applied for solving problems such as optimization problems, tangent problems, etc. This really should not be surprising as differential calculus literally is a set of rules for calculating differences. These rules first appeared in Leibniz’s 1684 paperNova methodus pro maximus et minimus, itemque tangentibus, quae nec fractus nec irrationals, quantitates moratur, et singulare pro illi calculi genus(A New Method for Maxima and Minima as Well as Tangents, Which is Impeded Neither by Fractional Nor by Irrational Quantities, and a Remarkable Type of Calculus for This)....

25. 22 An “Impossible” Problem, Courtesy of Leonhard Euler
(pp. 169-178)
Homer S. White

The second semester of calculus may well be the busiest course in the standard undergraduate mathematics curriculum. Between applications of integration, integration techniques, polar coordinates, parametric representations of curves, sequences and infinite series, one usually has no time to give conic sections their due. For quite some time, therefore, I have been looking for interesting things to say about conics that tie in well with students’ recently acquired calculus tools.

Recently I got lucky. I happened upon an article² published in 1755 by the great Swiss mathematician Leonhard Euler, which considers a problem that fits the bill perfectly. Euler’s treatment...

26. 23 Multiple Representations of Functions in the History of Mathematics
(pp. 179-188)
Robert Rogers

During the fall semester of 2005, I was slated to teach University Calculus I to a class of mostly incoming freshmen. It had been a while since I taught both the class and freshmen, so on the first day I decided to do some review and pick my students’ brains. I wrotey=f(x) =x2on the board and asked if that was a function. The unanimous answer was yes. Without exploring that too much, I drew Figure 23.1 on the board and asked if that was a function.

The response was overwhelmingly yes and when I asked...

27. 24 The Unity of all Science: Karl Pearson, the Mean and the Standard Deviation
(pp. 189-198)
Joe Albree

In statistics, Karl Pearson’s (1857–1936)method of momentsunified the arithmetic mean, the standard deviation, and a number of further statistical calculations. It may be surprising to learn that the underlying concepts of the method of moments come from physics. For Karl Pearson, though, this development was a natural one.

After introducing the story of Karl Pearson’s journey to the study of statistics, we present a set of practical data values which can be analyzed directly or grouped into classes and then analyzed. As we obtain the mean and standard deviation of this data set, we will see how...

28. 25 Finding the Greatest Common Divisor
(pp. 199-202)
J.J. Tattersall

One of the more important mathematical concepts students encounter is that of the greatest common divisor (gcd), the greatest positive integer that divides two integers. It can be used to solve indeterminate equations, compare ratios, construct continued fraction expansions, and in Sturm’s method to determine the number of real roots of a polynomial. For a development of these applications, see [1]. Most of the significant applications of the gcd require that it be expressed as a linear combination of the two given integers. The gcd and its associated linear equation provide an efficient way to find inverses of elements in...

29. 26 Two-Way Numbers and an Alternate Technique for Multiplying Two Numbers
(pp. 203-208)
J.J. Tattersall

In 1726, John Colson (1680–1759), a British mathematician and member of the Royal Society of London, devised an ingenious way to represent positive integers using what he called negativo-affirmative figures.[2] With his scheme positive and negative digits are intermingled and the basic arithmetic operations of addition, subtraction, and multiplication are as straightforward as in decimal arithmetic. The figures can be used to encrypt integers and have been rediscovered on several occasions. One version makes unnecessary the use of the digits 6, 7, 8,and 9, another rotates the digits 180°. Colson referred to his method as a “promiscuous scheme” to...

30. 27 The Origins of Integrating Factors
(pp. 209-214)
Dick Jardine

In a differential equations course, students learn to use integrating factors to solve first order linear differential equations, and in the process reinforce learning of key concepts from their calculus courses. This capsule offers some differential equations solved by the originators of the technique of using an integrating factor, though they did not use that expression. Solving differential equations via integrating factors is difficult for some students, particularly those who try to memorize a formula. We advocate that students learn to derive the method and solve differential equations using the product rule and the fundamental theorem of calculus, as advocated...

31. 28 Euler’s Method in Euler’s Words
(pp. 215-222)
Dick Jardine

Euler’s method is a technique for finding approximate solutions to differential equations addressed in a number of undergraduate mathematics courses. Various current texts include Euler’s method for calculus [4], differential equations [1], mathematical modeling [9], and numerical methods [2] students. Each of those courses are opportunities to give students an opportunity to read Euler’s own brief description of the algorithm, and in the process come to understand the technique and its limitations from Euler himself. This capsule includes historical information about Euler and his development of the approximation method. Additionally, I describe Student Assignments (Appendix A) I use to connect...

32. 29 Newton’s Differential Equation $\frac{{\dot{y}}}{{\dot{x}}}=1-3x+y+xx+xy$
(pp. 223-228)
Hüseyin Koçak

In this note we redress Newton’s solution to his differential equation in the title above in a contemporary setting. We resurrect Newton’s algorithmic series method for developing solutions of differential equations term-by-term. We provide computer simulations of his solution and suggest further explorations.

The only requisite mathematical apparatus herein is the knowledge of integration of polynomials. Therefore, this note can be used in a calculus course or a first course on differential equations. Indeed, the author used the content of this paper while covering the method of series solutions in an elementary course in differential equations. Additional specific examples studied...

33. 30 Roots, Rocks, and Newton-Raphson Algorithms for Approximating $\sqrt {2}$ 3000 Years Apart
(pp. 229-240)
Clemency Montelle

One of the classic examples to demonstrate the so-called Newton-Raphson method in undergraduate calculus is to apply it to the second-degree polynomial equationx2− 2 = 0 to find an approximation to the square-root of two. After several iterations the solution converges quite quickly. Indeed,$\sqrt {2}$has fascinated mathematicians since ancient times, and one of its earliest expressions is found on a cuneiform tablet written, it is supposed, some time in the first third of the second millennium b.c.e by a trainee scribe in southern Mesopotamia. While keeping this mathematical artifact firmly in its original archaeological and mathematical context,...

34. 31 Plimpton 322: The Pythagorean Theorem, More than a Thousand Years before Pythagoras
(pp. 241-250)
Daniel E. Otero

An amazingly sophisticated example of some of the oldest written mathematics known to humanity is the clay tablet Plimpton 322 (Figure 31.1), so called because it is item number 322 in a collection assembled by G. A. Plimpton in the 1930s and now housed at Columbia University in New York City. The tablet dates to the 19th century BCE, and can be traced to the Old Babylonian civilization that flourished in Mesopotamia, the fertile valley of the Tigris and Euphrates rivers (present-day Iraq). This exotic artifact is an ideal touchstone that can be used to spark interest in the study...

35. 32 Thomas Harriot’s Pythagorean Triples: Could He List Them All?
(pp. 251-260)
Janet L. Beery

English mathematician and scientist Thomas Harriot (1560–1621) gave the usual formula for Pythagorean triples using his new algebraic notation but he also started to list them in a systematic way. If he could have continued his list indefinitely, would he have listed all of the Pythagorean triples? In exploring this question, students can recognize and describe patterns, write and use algebraic formulas, and construct proofs, including proofs by mathematical induction. Students also have the opportunity to study an historical approach in which a mathematician seemed to believe that tabular presentation of a result was just as valuable, effective, and...

36. 33 Amo, Amas, Amat! What’s the sum of that? Bernoulli’s Account of the the Divergent Harmonic Series in Latin
(pp. 261-268)
Clemency Montelle

Every course in undergraduate calculus contains some component of the examination of series and the various tests to establish their convergence. One of the most important series is the Harmonic series, which is not only mathematically interesting per se, but also appears frequently as an ideal ‘comparison’ series to determine the convergence or divergence of other series. At some point, the formal proof of its divergence must be covered. This paper provides a quirky alternative to the format and the content of the standard proof usually offered; a capsule based on an examination of the actual primary source of the...

37. 34 The Harmonic Series: A Primer
(pp. 269-276)

Students in a first course of real analysis are often bewildered by many things, but perhaps the main difficulties they encounter are centered around three fundamental concepts: the notion of infinity and infinite processes; the phenomena of convergence and divergence; and the construction of rigorous proofs. It is in just such a course that historical information can be used to good effect. After all, the fact that these are issues with which mathematicians have grappled for centuries will doubtless be of some reassurance to the student struggling to master them.

What may be less comforting (but important, nevertheless, for a...

38. 35 Learning to Move with Dedekind
(pp. 277-284)
Fernando Q. Gouvêa

The history of mathematics sometimes calls our attention to intellectual hurdles that our students must face, showing that ideas and conceptual moves that have become second nature to us are in fact quite daring and difficult to learn. This article focuses on a particular conceptual move, which we call “the Dedekind move” because it was so characteristic of Richard Dedekind’s work. Briefly put, the idea isto define a mathematical object as a set of other mathematical objects. We then treat the whole set as a single thing, and do our best to forget its original plural nature.

Students typically...