Introduction

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 student to know) is that while mathematical results, once proved, are permanent, the arguments on which they are based are sometimes less so. In other words, as mathematics has developed, so has the concept of mathematical rigor. The result is that, although a theorem first proved 250 years ago is still true, the proof originally given for it might not be regarded as fully satisfactory by today's mathematicians. Given that the result still holds, however, it is not normally too difficult to formulate an alternative proof that attains modern standards of mathematical rigor.

A good example is the subject of this chapter, the harmonic series, which provides both a simple and an important introduction to issues surrounding the convergence and divergence of infinite series. Moreover, by virtue of its rich history and the related mathematical topics that arise from its study, it is a particularly appropriate tool to use when introducing students to the rudiments of classical analysis.

Introduction

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, or in library books on the history of mathematics, but the activities in this article are designed so that students are encouraged to search the Internet for information. In the process, students will probably be surprised to learn how much material is ‘out there’ about mathematics and its history and will begin to learn how to separate the online wheat from the chaff. The examples that follow are intended for students in linear algebra and differential equations courses, as indicated, but you can obviously use this idea in any class. This assignment is very flexible: you may assign the worksheets for homework or use them as a class activity; you may have students work independently or in groups.

Students enrolled in a linear algebra class may never have stopped to think that Gaussian elimination was named for someone named Gauss, or that there was a Cramer behind Cramer's Rule. They may be surprised to learn that people have been solving systems of linear equations for thousands of years.

Introduction

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.

Historical Background

In La Géometrie (1637) [2] Renée Descartes presents his general method of drawing a straight line to make right angles with a curve at an arbitrarily chosen point upon it. He praises his own approach as solving “not only the most useful and most general problem in geometry that I know, but even that I have ever desired to know” [2, p. 95]. In our terms, he sought the normal line to a curve at a given point, from which the tangent line can easily be found as well.

Descartes' approach is quite different from the modern one, which raises the question: If his method was as important as he thought,why did it not prevail? I will describe his method and suggest some exercises that can help a student to understand what Descartes was doing and to see why other approaches won out. Using calculators or computer algebra systems allows us to remove much of the drudgery of this historical reenactment.

In the Classroom

Descartes' Approach to Tangents

Figure 13.1 is based very loosely on Descartes' own diagram [2, pp. 94 and 98]. The most important anachronismis the vertical axis.

Introduction

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 algorithm from their mathematical toolkits whenever needed and deploy it as a rote arithmetical procedure.

I have adapted for instructional use a North African version of the rule of double false position. The topic is well suited to college or high school courses in College Algebra, Precalculus, Calculus, Applied Calculus, and Linear Algebra. In my experience, only 30–50 minutes of class time needs to be devoted to teaching the method in order for students to grasp the mechanics, justification, and various applications. Instruction can take any of various forms, ranging from a traditional lecture to a self-guided instructional module for individual or group work. I describe such a module below, in the section “In the Classroom”.

Covering a technique that students will find handy in solving certain problems helps round out their technical skills. In addition, it helps introduce them to the contributions of a variety of cultures, and provides some historical perspective on mathematics.

Introduction

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 proof, as it originally appeared, in Latin.

This capsule should ideally be offered before covering the various convergence tests, and just after examining geometric series. It could be particularly fitting to include it as part of your coverage of the divergence test as the Harmonic series is often the example cited to demonstrate that the convergence of terms in a series that tend to zero is not sufficient to guarantee the convergence of the actual series.

Given the richness of historical insight, the relevance of the mathematics, and indeed the novelty for the students, the presentation of this primary source is ideal for the undergraduate mathematics classroom. Grabbing the attention of the students by presenting something completely different, yet utterly relevant, may very well renew their enthusiasm as well as stimulating curiosity and assisting their grasp of this topic.

Introduction

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 in x and y — 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 geometry, precalculus or calculus. For students who know some calculus, it also provides insight into continuous nondifferentiable functions. There is also a connection to power series representations, although this will not be discussed in this chapter (Euler treats them in §5–9 of [1, 2]).

For further reading on these topics, see [3, 4].

Historical Background

In the 18th century, calculus and the related branches of mathematics gradually changed their perspective from the geometric to the algebraic. When Renée Descartes (1596–1650) and Pierre de Fermat (1601–1665) invented analytic geometry, for example, mathematicians were already familiar with a large assortment of curves, given by a variety of geometric constructions. Analytic geometry gave them a means of associating equations with these curves. With passing time, the study of equations took primacy, so that the graph came to be seen as an attribute of the equation.

Introduction

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 of representations of number and of arithmetical computational algorithms, say, by future computer scientists or prospective school teachers. It can also serve to deepen an understanding of the solution of quadratic equations by students of algebra at all levels.

Historical Background

Evidence of mathematical thinking is at least as old as the species homo sapiens. Older, in fact, provided we agree to classify certain animal behaviors, like the ability to differentiate quantities, to count, or even to employ geometric design in the building of shelters, as evidence of mathematical activity.

Once humans moved from hunting and foraging to farming and later to forming cities, new challenges of life required new forms of thought, including those that looked much more like what we would today identify as mathematics. Certainly, the first literate civilizations known to us also provided written evidence of their mathematics as well.

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 is to 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 meet this idea for the first time in an “introduction to abstraction” class, when they learn about equivalence classes. It really comes into its own, however, when the quotient construction is introduced in abstract algebra. This is a notorious stumbling block for students. A little history can help us understand why, and suggests some ideas for helping students over the hump.

Historical Background: What Dedekind Did

Suppose we are confronted with the need to come up with a definition of some mathematical entity. There are many ways to go about this. Some mathematical definitions, for example, are entirely functional: we explain what it does, and ignore completely the issue of what it is. But this is rarely completely satisfying. How do we even know that the object in question exists? Some construction is usually wanted.

Richard Dedekind was faced with this situation more than once. His approach was fairly consistent.

Introduction

Nowadays we recognize written algebra by the presence of letters (called variables) standing for unspecified numbers, and especially by the presence of equations involving those letters. These two features—letters and equations—reveal the techniques of algebra, but algebra itself is not these 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 must perform (multiplication followed by addition). But if you were asked for a number having the property that if it is multiplied by 3 and 26 is added to the product, the result is 57, you would be facing an algebra problem. In an algebra problem, the operations and some of the data are given to you, but these operations are not for you to perform. Rather, you assume someone else has performed them, and you need to find the number(s) on which they were performed. Using this definition, we can recognize algebra problems in very ancient texts that contain no equations at all.

But how do such problems arise?Why were people interested in solving them? Those are questions that any student who looks beyond the horizon of tomorrow's homework assignment is bound to ask. In the following paragraphs, we shall look at some examples and see if we can answer such questions.

Introduction

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 equation x2 – 2 = 0 to find an approximation to the square-root of two. After several iterations the solution converges quite quickly. Indeed, √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, we look at the similarities and differences it shares with modern mathematical techniques, 3000 years distant.

Observing that mathematical knowledge is, to a certain extent, culturally dependent can be revelatory to students. Modern mathematical pedagogy is generally based around a cumulative approach which allows little room for lateral breadth, as it focuses on the acquisition of skills, often with scant regard for the actual manifestation or circumstances of mathematical knowledge itself. Students may have never been exposed to other contexts in which mathematics flourished, nor encountered different mathematical traditions thus far in their studies, much less non-western ones. Yet, such exposure can give them a vital and nuanced perspective on their own mathematical circumstances. Though many a mathematical problem posed may be universal, the ways in which various mathematically literate cultures attacked them and solved them are diverse, depending on many other factors related to the broader intellectual environment. This is an important observation to bequeath to future mathematically-literate generations. However, at the same time, the universality of mathematics should not be forgotten.

Mathematical Time Capsules offers teachers historical modules for immediate use in the mathematics classroom. Relevant history-based activities for a wide range of undergraduate and secondary mathematics courses are included. The genesis of this volume was a Contributed Papers Session on Using History of Mathematics in Your Mathematics Courses, organized by the editors at the Joint Mathematics Meetings, San Antonio, Texas, in January of 2006. That session was very well attended, which prompted Andrew Sterrett from MAA publications to suggest that we put together our second volume for the MAA Notes series.

Purpose

For a wide variety of reasons, instructors are looking for ways to include the history of mathematics in their courses. It is not uncommon to see requests for “how to” posted to the History of Mathematics Special Interest Group of the MAA (www.homsigmaa.org) email list, such as this 2008 posting:

In response to such inquiries, we hope to serve the broader mathematical community by offering practical suggestions on how to use the history of mathematics quickly and easily in the mathematics classroom.

A time capsule can be defined as a container preserving articles and records from the past for scholars of the future. Of course our volume does not fit that precise definition, but readers who open this book will find articles and activities from mathematics history that enhance the learning of topics typically associated with undergraduate or secondary mathematics curricula.

Introduction

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 that history to the mathematics the students are learning. The activities are designed to deepen student understanding of Euler's method specifically and reinforce learning of calculus skills in general. I also include a translation of Euler's writing on the topic (Appendix B).

Historical preliminaries

Leonhard Euler (1707–1783) was one of the most gifted of all mathematicians. Excellent biographies of Euler, some identifying the voluminous quantity of his mathematical writing, are available [6], which interested readers are encouraged to explore. One of Euler's many gifts was his ability to write mathematics clearly and understandably.

Introduction

In statistics, Karl Pearson's (1857–1936) method of moments unified 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 the physics of the first and second moments aid in our computations, and of even more importance, give us insights into the results of the calculations.

Karl Pearson: Historical preliminaries

Even though Karl Pearson has been heralded as “the founder of the twentieth century science of statistics” [2, p. 447], his story had a much different beginning. Carl Pearson (as he was christened) grew up in an upper middle class Victorian London home. In 1875, Carl earned a scholarship at King's College, Cambridge, where he studied the works of Charles Darwin (1809–1882) and of Benedict Spinoza (1632–1677), and German history. He graduated with honors in mathematics (1879). After graduation, Pearson traveled and studied in Germany, where he became so enamored with the works of Karl Marx that he changed the legal spelling of his name from Carl to Karl; to his friends and colleagues, he was also known as K.P. When he returned to London, he was admitted to the bar (1881), and as his father wished, he practiced law for a short time.

Introduction

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 article published in 1755 by the great Swiss mathematician Leonhard Euler, which considers a problem that fits the bill perfectly. Euler's treatment of the problem synthesizes a number of ideas from elementary calculus: trigonometric identities, techniques of integration including partial fractions, representation of curves by polar equations, and separable differential equations, with a particular conic section—the parabola-leading off the action.

Historical Setting

Suppose that you are given a parabola, and that you draw an arbitrary line through its focus F, meeting the parabola at points M and N. The tangent lines to these points will always meet at a right angle! One possible approach to a proof is to work from the reflection property of parabolas, as follows:

In Figure 22.1, the points P and Q are chosen so that PN and QM are parallel to the central axis of the parabola. By the reflection property, a ray of light traveling from P to N will bounce off the parabola and head toward F, with PN and NF making equal angles, of measure α let us say, to the tangent line YZ at point N.

Introduction

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 by the luminaries in the early history of differential equations are available in [1]. This work was supported by the National Science Foundation's Course, Curriculum, and Laboratory Improvement Program under grant DUE-0230612

Newton's differential equation

Newton's book [6], ANALYSIS Per Quantitatum, SERIES, FLUXIONES, AC DIFFERENTIAS: cum Enumeratione Linearum TERTII ORDINIS consists of one dozen problems. The second problem

is devoted to a general method of finding the solution of an initial-value problem for a scalar ordinary differential equation in terms of series. The equation in the title of the present paper (see also Figure 29.1) is the first significant example in the section on PROB. II.

Newton thought of mathematical quantities as being generated by a continuous motion.

Introduction

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 print, 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's Elements? In this proof, Euclid introduces a clever and elegant application of the concept of shearing. It is a proof that provides a golden opportunity not only to bring some history into the classroom, but that also provides us a natural venue to highlight connections between algebraic and geometric concepts. Moreover, the proof presented by Euclid has the useful property that it provides for generalizations of the theorem in a number of different directions. For example, by using shearing one may prove Pappus' theorem, which is a Pythagorean-like theorem for arbitrary triangles. The concept of shearing itself can then be generalized in the form of Cavalieri's principle to determine the volume of more general solids.

In Book XII, Euclid again applies a technique that is connected to the concept of shearing, this time in three dimensions. The problem is seemingly unrelated: determining the relationship between the volumes of pyramids and prisms that share the same base and height. This application provides contemporary teachers an opportunity to motivate and illuminate the ostensibly nonintuitive formula for the volume of a pyramid.

Introduction

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 when the students have an opportunity to actively engage in an assignment that builds on prior knowledge. The debate assignment was designed to do just that — build upon and shore up the students' understanding of the key concepts of first semester calculus. Bonwell and Eison [1] explain that students are actively learning when they are asked not just to listen, but also to analyze, to synthesize, and to evaluate through active engagement. The debate project requires students to create and analyze arguments, and to actively present these arguments during an in-class debate. As the debate assignment progresses, students begin to take ownership of their arguments and are concerned that they present these arguments well, spending a surprising amount of time in preparation for the debates. During their preparation for and participation in the debates, the students gain a better understanding of some of the fundamental aspects of their beginning calculus course and thus are more likely to remember these fundamental ideas.

Introduction

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 Latin quadratum, 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 wide range of reasons including that it provides arguably the best approach to deriving the quadratic formula. In the examples below, we outline the use of completing the square as it was done in four previous millennia.

Over the years, the method has had various representations. Understanding the historical, geometric representation may help students internalize the method when algorithmic or algebraic representations alone may not. Multipleways of learning and knowing are offered by including the historical perspective. The examples given in this capsule are actual problems solved in the past, and your students are invited to solve them today using the methods of antiquity as well as current techniques. In my courses, I present the information as an interactive lecture that extensively involves students, as described below.

Historical preliminaries

About 4000 years ago, Mesopotamian scribes pressed the method of completing the square into clay tablets, the technology used to record information in that time.