# A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources

Author(s):
Janet Heine Barnett (Colorado State University - Pueblo), Kathleen M. Clark (Florida State University), Dominic Klyve (Central Washington University), Jerry Lodder (New Mexico State University), Daniel E. Otero (Xavier University), Nick Scoville (Ursinus College), Diana White (University of Colorado Denver)

Regular readers of Convergence are unlikely to need to be convinced of the value of studying primary sources in order to understand the development of mathematics. Essentially all good scholarship in mathematics is necessarily based on primary sources. But, as many have come to recognize, primary source materials are also valuable in learning the subject.  Thanks in part to the work of David Pengelley [11], Jerry Lodder [10], Janet Barnett [3], and others [1,2,5,6,12], ideas about the incorporation of primary sources in the teaching of university mathematics have become increasingly well-known over the last few decades. Dozens of mathematics faculty have given talks recently at national meetings about this pedagogical innovation, and several articles in Convergence have described details of such an approach [7,8,9,13,14], and even provided collections of classroom-ready student projects based on primary sources [4].

An outside observer, seeing this significant increase in the attention paid to the role of primary sources in teaching, may think that the movement to promote this method of teaching has been successful, and that these techniques are now widely used.

Some of us, however, share the view that the techniques are not nearly mainstream enough. That faculty with an interest in primary sources can use them in their teaching is well and good, but we are convinced that there are so many benefits derived from their use that we would like to see them available to all instructors of university mathematics.

Students work on a Primary Source Project under the supervision of Janet Barnett at a TRIUMPHS Site Tester Workshop, held at the University of Colorado, Denver, during September of 2016.

Five years ago, at a workshop for the NSF-funded project Learning Discrete Mathematics and Computer Science via Primary Historical Sources in Las Cruces, New Mexico, a group of us started planning a large project: a national, funded, extended effort to build on the successful work done over the previous decades to promote this work of teaching with primary sources and help to make it more common. We would do this by continuing to prepare ready-to-use curriculum materials in the form of classroom projects based on primary source materials through which students could learn a wide variety of standard topics in university mathematics classes. If more such projects were available, we reasoned, they might significantly reduce the perceived difficulty some instructors may have in using original sources for the first time, especially if this broader collection of projects included a variety of shorter projects. Furthermore, if we combined the work of disseminating these projects with a careful research plan, we might learn something important about the benefits or barriers of using such an approach.

In 2015, our group, which had by this time expanded to a team of seven, secured a Collaborative Research grant from the National Science Foundation of about \$1.5 million over five years to run this grand experiment on a large scale. We have been working actively since then first to write and edit these “Primary Source Projects”, or PSPs, then to test them in classrooms – either our own or in those of colleagues at other institutions.

The grant effort, TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (or TRIUMPHS), is well underway, and we encourage readers to find out more about its current status at the TRIUMPHS website. In addition to administrative notices governing the work of the grant, we maintain there an updated set of PSPs, available for perusal and download.

We’d especially like to call attention to a new set of "mini-PSPs" which takes our work in an exciting new direction. These shorter Primary Source Projects are designed to be completed within one or two class periods, and to give instructors already familiar with primary sources more flexibility in their use. We plan to publish a series of these mini-PSPs in Convergence over the next few years - beginning with the mini-PSP The Derivatives of the Sine and Cosine Functions - and we invite the mathematical community to read and use them.

If readers do decide to use projects in their classes, of course, we would love to know about it – contact the project’s author or any member of the TRIUMPHS team in advance, and we’ll be happy to talk with you. And have fun exploring them!

Subset of TRIUMPHS PIs and Advisory Board Members

Fom left to right: David Pengelley, Diana White, Danny Otero, Dominic Klyve, Janet Barnett, Kathy Clark, Nick Scoville

##### Acknowledgements

This material is based upon work supported in part by the National Science Foundation under Grants No. 1523494, 1523561, 1523747, 1523753, 1523898, 1524065, and 1524098. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

##### References
1. J. Barnett, G. Bezhanishvili, J. Lodder, and D. Pengelley, Teaching Discrete Mathematics Entirely From Primary Historical Sources, PRIMUS: Problems, Resources and Issues in Mathematics Undergraduate Studies, 26 (2016) no. 7, 657-675, DOI 10.1080/10511970.2015.1128502.
2. J. Barnett, J. Lodder, and D. Pengelley, Teaching and Learning Mathematics From Primary Historical Sources, PRIMUS: Problems, Resources and Issues in Mathematics Undergraduate Studies, 26 (2016) no. 1, 1 - 18, DOI 10.1080/10511970.2015.1054010.
3. J. Barnett, Learning mathematics via primary historical sources: Straight from the source’s mouth, PRIMUS: Problems, Resources and Issues in Mathematics Undergraduate Studies: Special Issue on the Use of History of Mathematics to Enhance Undergraduate Mathematics Instruction, 24 (2014), no. 8, 722–736, DOI:10.1080/10511970.2014.899532.
4. J. Barnett, G. Bezhanishvili, H. Leung, J. Lodder, D. Pengelley, I. Pivkina, D. Ranjan, and M. Zack, Primary Historical Sources in the Classroom: Discrete Mathematics and Computer Science, Loci: Convergence (July 2013), DOI 10.4169/loci003984.
5. J. Barnett, H. Leung, J. Lodder, D. Pengelley, and D. Ranjan, Designing Student Projects for Teaching and Learning Discrete Mathematics and Computer Science via Primary Historical Sources, in Recent Developments on Introducing a Historical Dimension in Mathematics Education (V. Katz and C. Tzanakis, eds.), Mathematical Association of America, Washington DC, 2011,  187–198.
6. J. Barnett, J. Lodder, and D. Pengelley, The pedagogy of primary historical sources in mathematics: Classroom practice meets theoretical frameworks, Science & Education: Special Issue on the Philosophy and History of Mathematics in Mathematics Education 23 (2014), no. 1, 7–27.
7. K. M. Clark, 'In these numbers we use no fractions': A Classroom Module on Stevin's Decimal Fractions - Overview and Introduction, Loci: Convergence (January 2011), DOI:10.4169/loci003333.
8. A. Dematte, Introducing the History of Mathematics: An Italian Experience Using Original Documents, Loci: Convergence (February 2010), DOI:10.4169/loci002856
9. D. Klyve, L. Stemkowski, and E. Tou, Teaching and Research with Original Sources from the Euler Archive, Loci: Convergence (April 2011), DOI 10.4169/loci003672.
10. J. Lodder, Networks and spanning trees: the juxtaposition of Prüfer and Borůvka. PRIMUS: Problems, Resources and Issues in Mathematics Undergraduate Studies: Special Issue on the Use of History of Mathematics to Enhance Undergraduate Mathematics Instruction, 26 (2016) 24 (2014), no. 8, 737 -751, DOI: 10.1080/10511970.2014.896835.
11. D. Pengelley, Teaching with primary historical sources: Should it go mainstream? Can it?, in Recent Developments on Introducing a Historical Dimension in Mathematics Education (V. Katz and C. Tzanakis, eds.), Mathematical Association of America, Washington DC, 2011, 1–8.
12. D. Ruch, Creating a Project on Difference Equations with Primary Sources: Challenges and Opportunities, PRIMUS: Problems, Resources and Issues in Mathematics Undergraduate Studies: Special Issue on the Use of History of Mathematics to Enhance Undergraduate Mathematics Instruction, 24 (2014), no. 8, 764–773, DOI: 10.1080/10511970.2014.896836.
13. N. Scoville, Georg Cantor at the Dawn of Point-Set Topology, Loci: Convergence (March 2012), DOI: 10.4169/loci003861.
14. L. Stemkoski, Investigating Euler's Polyhedral Formula Using Original Sources - Original Sources: Who, What, Where, How, Convergence (April 2010), DOI:10.4169/loci003297.

# The Derivatives of the Sine and Cosine Functions: A Mini-Primary Source Project for Calculus I Students

Author(s):
Dominic Klyve (Central Washington University)

This curricular module guides students through a method of calculating the derivative of the sine and cosine functions using differentials. It is based on one primary source: Leonhard Euler's Institutiones calculi differentialis (Foundations of Differential Calculus) [2], published in 1755. It was the first calculus book to use functions; indeed, Euler himself had been the first mathematician to regularly use an approach which looks like functions to us today about seven years earlier, in his great "pre-calculus'' book, the Introductio in analysin infinitorum (Introduction to the Analysis of the Infinite) [3]. While the use of functions makes the material more accessible to our students today, Euler's approach is different enough from that of modern calculus books that it forces students to think carefully about the material.

The major difference in Euler's approach is the lack of limits in his work. The limit concept would not be formally defined and made a part of mathematics for almost a century; Euler based his calculus (following Leibniz) on the differential dx, which was an infinitely small increment of the variable x. While the logical issues in this approach would force 19th-century mathematicians to abandon it (in favor of limits), Euler saw no such issues. (It is the author's experience that Calculus students are closer to Euler on the epistemological spectrum concerning differentials, and are happy to use them for insight; the author further believes that they provide this insight. Fortunately, this belief will be tested by TRIUMPHS' research component!)

Perhaps the most surprising aspect of Euler's approach is his use of Taylor series. In fact, he introduced these in the Introductio, and thought of them as a pre-calculus idea. In this way, Euler presaged the later work of Lagrange [4], who used power series as the starting point for his theory of calculus, and defined the derivative as the “first derived function” in the power series expansion of any given function.

This project may be the first time that students see these series, but they do not need any of the theory of Taylor series in the project. The approximations of sine and cosine via three-term Taylor series are presented as a fait accompli, and students are given an opportunity to convince themselves that the approximation seems valid, even if they can't explain why. It is hoped that this exposure will make Taylor series slightly more approachable when they encounter them in the future, but this is not a major goal of the project.

The project The Derivatives of the Sine and Cosine Functions (pdf file) is ready for student use, and the LaTeX source code is available from the author by request.

At the end of the project is a set of instructor notes. These echo some of the material in this introduction, and also include practical advice for the use of the project in the classroom.

This project is the first in A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources that is planned for publication in Convergence, for use in courses ranging from first year calculus to analysis, number theory to topology, and more.  Links to other mini-PSPs in the series appear below.

Acknowledgments

The development of the student project The Derivatives of the Sine and Cosine Functions has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project with funding from the National Science Foundation’s Improving Undergraduate STEM Education Program under Grants No. 1523494, 1523561, 1523747, 1523753, 1523898, 1524065, and 1524098. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily reflect the views of the National Science Foundation.

References

[1] Edwards, Harold M. (2007). Euler's definition of the derivative. Bulletin of the American Mathematical Society. 44 (4): 575–580.

[2] Euler, Leonhard (1755). Institutiones Calculi Differentialis, St. Petersburg. Translation by John D. Blanton in Foundations of Differential Calculus, Springer, New York (2000).

[3] Euler, Leonhard (1748). Introductio in Analysin Infinitorum, St. Petersburg. Translation by John D. Blanton in Introduction to the Analysis of the Infinite, Springer, New York (1988).

[4] Lagrange, Joseph-Louis (1797). Théorie des fonctions analytiques, Paris.

# Why be so Critical? Nineteenth-Century Mathematics and the Origins of Analysis: A Mini-Primary Source Project for Introductory Analysis Students

Author(s):
Janet Heine Barnett (Colorado State University - Pueblo)

The seventeenth century witnessed the development of calculus as the study of geometric curves in the hands of Newton, Leibniz and their immediate followers, with algebra (or 'analysis') serving as an aid to that work. This situation changed dramatically in the eighteenth century when the focus of calculus shifted instead to the study of functions, a change due largely to the influence of Euler. In the hands of Euler and his contemporaries, functions became a powerful problem solving and modelling tool in physics, astronomy, and related mathematical fields such as differential equations and the calculus of variations. Soon thereafter, mathematicians began to express concerns about calculus (analysis) and its foundations. The language, techniques and theorems that developed in response to these concerns are precisely those which students encounter in an introductory analysis course – but without the context that motivated nineteenth century analysts.

This mini-Primary Source Project (mini-PSP) employs excerpts from the works of nineteenth century mathematicians as a means to introduce students to that larger context. By offering a glimpse into the problems that motivated mathematicians to shift towards a more formal and abstract study of the concepts underlying various calculus procedures and applications, the project supports students' success in making a similar shift in their own understanding of these concepts. Completing this mini-PSP early in the course can also provide students and instructors with a basis for reflection on and discussion of current standards of proof and rigor throughout the course.

 Bernard Bolzano (1781-1848) (Wikimedia Commons) Augustin-Louis Cauchy (1789-1857) (Wikimedia Commons) Richard Dedekind (1831-1916) (Wikimedia Commons)

The project begins with excerpts from the writing of Bernard Bolzano [2], Augustin-Louis Cauchy [3,4] and Richard Dedekind [5] in which these mathematicians described their various concerns about the state of analysis. Those concerns included, for example, a growing mistrust of geometric intuition as a valid method of proof for analytic truths. Project questions in this first section of the mini-PSP direct students' attention towards certain specific aspects of the quoted excerpts, and prompt them to compare and contrast the concerns of these three individuals.

 In the second section of the project, students examine an extended excerpt from a letter from Niels Abel to his high school teacher, Bernt Michael Holmboe, written in 1826 while Abel was living in Paris. This portion of the mini-PSP again includes questions that prompt students to reflect upon the general nature of the concerns Abel expressed about the state of mathematics at the time, as well as how his concerns relate to those expressed by the authors of the earlier excerpts. Other questions in this section lead students to explore the mathematical details of examples cited by Abel as just cause for his complaints about the lack of firm foundations for power series techniques in particular.   Divergent series are on the whole devilish, and it is a shame that one dares to base any demonstration on them. One can obtain whatever one wants, when one uses them. It is they which have created so much disaster and so many paradoxes. (Excerpted from [1]) Niels Abel (1802-1829) (Wikimedia Commons)

The complete project Why be so Critical? is ready for student use, and the LaTeX source code is available from the author by request.

A set of instructor notes offering practical advice for the use of the project in the classroom is appended at the end of the student project; these include suggested Summary Discussion Notes that can be used to guide a whole class discussion of the main project themes.

This project is the second in A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources that is planned for publication in Convergence, for use in courses ranging from first year calculus to analysis, number theory to topology, and more. Links to other mini-PSPs in the series appear below. The full TRIUMPHS collection includes eleven PSPs for use in a real analysis course.

Acknowledgments

The development of the student project Why be so critical? has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project with funding from the National Science Foundation’s Improving Undergraduate STEM Education Program under Grant No. 1523494. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily reflect the views of the National Science Foundation.

References

[1] Abel, N., "Breve fra og til Abel," in Festskrift ved Hundredeaarsjubilæet for Niels Henrik Abels Fødsel (editors E. Holst, C. Stømer and L. Sylow), Kristiana: Jacob Dybwad, 1902.

[2] Bolzano, B., Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetzes Resultat gewähren, wenigstens eine reele Wurzel der Gleichung liege, Leipzig: W. Engelmann, 1817.

[3] Cauchy, A., Cours d'Analyse de L'Ecole Royale Polytechnique, Paris: Debure, 1821.

[4] Cauchy, A., Résumé Leçons sur le calcul infinitésimal, Paris: Debure, 1823.

[5] Dedekind, R., Stetigkeit und irrationale Zahlen, Braunschweig: F. Vieweg und Sohn, 1872.

# Connecting Connectedness: A Mini-Primary Source Project for Topology Students

Author(s):
Nicholas A. Scoville (Ursinus College)

One of the main obstacles for students seeking to understand higher mathematics is the need to grasp a definition. As professional mathematicians, we know that definitions do not fall from the sky, but are instead arrived at through careful and painstaking work. When all is said and done, a good definition is pithy and precise, having been carefully molded with all the right nuances and wording to include the cases we want and to exclude pathologies.

It would seem, however, that this must be news to our students, as the learning of mathematics often begins with a definition rather than ending with it. One of the first places that this traditional pedagogy can begin to cause serious problems for the student is in a first course in topology. A topological space is already a fairly abstract concept, and the properties of topological spaces that one might be interested in studying are even more abstract.

One way to remedy this problem is to trace the evolution of a definition through its historical developments. Although not every definition in topology has a robust historical evolution, one definition with an especially fascinating history is that of connectedness. The concept of connectedness is also one of the more intuitive concepts that is encountered in topology, so that its evolution is quite remarkable to see. This mini-Primary Source Project (mini-PSP), Connecting Connectedness, is an all-too-brief sketch of that evolution.

 Connectedness was originally defined by Georg Cantor in 1883 as a condition that a set needed to satisfy in order to constitute a continuum [1]. Two things about Cantor’s definition are especially noteworthy. First, the definition presupposes the existence of a metric. Second, it applies only to closed and bounded sets. In particular, it made no sense for Cantor to ask if the punctured interval $$\, [0,1]-\{{1/2}\}$$ was connected. This example becomes a running theme throughout the project, with the question of whether a particular definition can determine that this set is not connected (under the usual topology in $${ R}$$, of course) serving as one criterion by which we might judge whether or not we have arrived at a fully satisfactory definition. Georg Cantor (1845-1918) (Wikimedia Commons)
 Camille Jordan (1838-1922) (Wikimedia Commons) The mini-PSP then looks at the work of Camille Jordan through excerpts from his well-known 1909 textbook Cours d’analyse [2]. While Jordan made several contributions to the evolving understanding of the concept of connectedness, the project itself focuses on a specific conceptual distinction that he identified in relation to the concept. Whereas Cantor defined a set to be connected if it satisfied a certain condition, Jordan defined a set to be separated if it satisfied a different condition, and then showed that a set is connected (in Cantor’s sense) if and only if there does not exist a separation. Even though Jordan did not give a new definition of connectedness, his viewpoint of separation as a property of connected sets is the one that we use today.
 The project’s next primary source comes from a 1904 work [4] by mathematician Arthur Moritz Schoenflies. Schoenflies had the insight that the concept of distance is not needed to define connectedness. By abstracting away the distance, he then defined the concept in purely point set terms. This was done, in part, to prove that connectedness is a topological invariant (an aspect of Schoenflies’ work that is not covered in the project). Arthur Schoenflies (1853-1928) (Convergence Portrait Gallery)

Finally, the project culminates with the 1911 work [3] of a Norwegian mathematician who spent his career at Montana State University, Nels Lennes. Lennes was attempting to prove the Jordan Curve Theorem and, in the process, gave the definition of connectedness that we use to this day. Rather than beginning their study of connectedness with the finalized form of the definition, students thus end their project work with that definition. We believe this pedagogy serves our primary goal of building more insight and intuition into the abstract notion of connectedness, as well as our secondary goal of offering students a glimpse into the rich and intriguing history of mathematics.

The complete project Connecting Connectedness (pdf file) is ready for student use, and the LaTeX source code is available from the author by request. A set of instructor notes that explain the purpose of the project and guide the instructor through the goals of each of the individual sections is appended at the end of the student project.

This project is the third in A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources that is planned for publication in Convergence, for use in courses ranging from first year calculus to analysis, number theory to topology, and more. Links to other mini-PSPs in the series appear below. The full TRIUMPHS collection includes five additional mini-PSPs and two more extensive “full-length” PSPs for use in a topology course, with one of these full PSPs telling the story of the evolution of “connectedness” in greater detail.

Acknowledgments

The development of the student project Connecting Connectedness has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project with funding from the National Science Foundation’s Improving Undergraduate STEM Education Program under Grants No. 1523494, 1523561, 1523747, 1523753, 1523898, 1524065, and 1524098. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily reflect the views of the National Science Foundation.

References

[1] G. Cantor, Uber unendliche, lineare Punktmannigfaltigkeiten 5, Math. Ann. 21 (1883), 545-586.

[2] C. Jordan, Cours d'analyse, vol. 1, 1893.

[3] N. J. Lennes, Curves in Non-Metrical Analysis Situs with an Application in the Calculus of Variations, Amer. J. Math. 33 (1911), no. 1-4, 287-326.

[4] A. Schoenflies, Beitrage zur Theorie der Punktmengen I, Math. Ann. 58 (1904), 195-238.

# Generating Pythagorean Triples: A Mini-Primary Source Project in Number Theory for Mathematics Majors, Elementary Teachers and Others

Author(s):
Janet Heine Barnett (Colorado State University - Pueblo)

In his commentary on Euclid's proof of the Pythagorean Theorem, Proclus (c. 411 - 485 CE) asserted (Proclus, pp. 339—340):

There are two sorts of right-angled triangles, isosceles and scalene. In isosceles triangles you cannot find numbers that fit the sides; for there is no square number that is the double of a square number, if you ignore approximations, such as the square of seven which lacks one of being double the square of five. But in scalene triangles it is possible to find such numbers, and it has been clearly shown that the square on the side subtending the right angle may be equal to the squares on the sides containing it. Such is the triangle in the Republic, in which sides of three and four contain the right angle and five subtends it, so that the square on five is equal to the squares on those sides. For this is twenty-five, and of those the square of three is nine and that of four sixteen. The statement, then, is clear for numbers.

Certain methods have been handed down for finding such triangles, one of them attributed to Plato, the other to Pythagoras.

Although their actual discovery is now hidden in the long shadow of history, the two methods for generating Pythagorean triples described by Proclus remain of interest in number theory today. Proclus’ simple description of how to generate Pythagorean triples using each method offered no speculation about how these algorithms may have been discovered. The mini-Primary Source Project Generating Pythagorean Triples provides students the opportunity to explore how these methods might have become known through an intriguing theory related to the Greek notion of a gnomon.

 With etymological roots in common with the English words gnostic, agnostic and ignorance, the literal meaning of the Greek work gnomon is "that which allows one to know.'' In astronomy, a gnomon is the part of a sundial that casts a shadow, thereby allowing one to know the time. In the ancient world, a vertical stick or pillar often served as the gnomon on a sundial. The term gnomon was also associated in ancient Greek architecture with an L-shaped instrument, sometimes called a 'set square,' that was used for the construction of (or 'knowing of') right angles. A hand-carved table-top analemmatic sundial. By John Carmichael (Own work) [Public domain], via Wikimedia Commons.

Within Greek mathematics, Euclid used the term gnomon to refer to the plane figure formed by removing any parallelogram from a corner of a larger similar parallelogram. Figure 1 shows the L-shaped gnomon associated with a square. Eventually, Greek mathematicians began to also use the term to refer to the increment between two successive figurate numbers. The square number gnomon diagram shown in Figure 2 is suggested in particular by certain passages in the Physics in which Aristotle (c. 384 - c. 322 BCE) discussed the mathematical beliefs of Pythagoras and his followers.

 Figure 1 Figure 2

The student project Generating Pythagorean Triples itself begins with an exploration of the primary source excerpt from Proclus above, followed by a brief introduction to the mathematical concept of a gnomon. It then returns to Proclus’ description of the two methods for generating Pythagorean triples which he attributed to Pythagoras and Plato respectively. For each of these two methods, students are presented with a series of tasks. These include numerical tasks based on Proclus' purely verbal descriptions of the two methods, algebraic tasks in which those verbal formulations are translated into symbolic formulas, and geometric tasks that connect the numerical and algebraic formulations of the method in question to gnomons in a figurate number diagram.

Two versions of the project are available, one somewhat more open-ended than the other. Beyond some basic arithmetic and (high school level) algebraic skills, no mathematical content pre-requisites are required in either version. This mini-PSP can thus be used with a wide range of students for whom the study of number theory is part of the curriculum.

Both versions of the project Generating Pythagorean Triples are ready for student use, subtitled The Methods of Pythagoras and of Plato via Gnomons (pdf file) and Gnomonic Explorations (pdf file) respectively. The latter version is the more open-ended, exploratory of the two versions. A set of instructor notes offering practical advice for the use of the project in the classroom is appended at the end of each version. These notes include more detail about the differences between the two versions of the project and the suitability of each for various student audiences. The LaTeX source code of the complete project is also available from the author by request.

This project is the fourth in A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources that is planned for publication in Convergence, for use in courses ranging from first year calculus to analysis, number theory to topology, and more. Links to other mini-PSPs in the series appear below. The full TRIUMPHS collection includes eight PSPs for use in courses on number theory.

Acknowledgments

The development of the student project Generating Pythagorean Triples has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project with funding from the National Science Foundation's Improving Undergraduate STEM Education Program under grant number 1523494. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily represent the views of the National Science Foundation. The author also wishes to thank George W. Heine III for creating the gnomon diagrams in Figures 1 and 2.

References

Proclus. A Commentary on the First Book of Euclid's Elements. English translation by Glenn R. Morrow (1970). Princeton: Princeton University Press.

# Euler's Rediscovery of e: A Mini-Primary Source Project for Introductory Analysis Students

Author(s):
Dave Ruch, Metropolitan State University of Denver
 Students meet the celebrated number $$e$$ in a number of places in their education, and at various levels of sophistication. This famous constant is used in countless applications and appears periodically in the evolution of mathematics. Leonhard Euler explored $$e$$ in his 1748 Introductio in Analysin Infinitorum [1], one of his most influential works.                                Jakob Emanuel Handmann's portrait of Euler (Wikimedia Commons) Title page of Introductio in Analysin Infinitorum (Convergence Mathematical Treasures Collection)

Euler wanted to examine exponential and logarithmic functions, especially as infinite series. Part of his challenge in working with logarithmic functions was to find a logarithmic base for which infinite series expansions are convenient. With this goal in mind, Euler derived $$e$$, both as the limiting value of the sequence $$\left( 1+1/j\right) ^{j}$$ and as the sum of the infinite series $$1+\dfrac{1}{1}+\dfrac{1}{1\cdot2}+\dfrac{1}{1\cdot2\cdot 3}+\dfrac{1}{1\cdot2\cdot3\cdot4}+\cdots.$$ Euler's remarkable development involves clever use of infinitely large and small numbers, which is an aesthetic treat but poses difficult questions for a modern student.

This mini-Primary Source Project, designed for an introductory course in analysis, is structured around passages from Euler's Introductio in Analysin Infinitorum. Students see how $$e$$ appears naturally in his development of exponential and logarithmic functions. A vital component of this project is giving a modern justification of $$e=\lim_{j\rightarrow\infty}\left( 1+1/j\right) ^{j}$$ using Euler's ideas along with some modern theory. The approach using the Monotone Convergence Theorem, as outlined in project exercises, is a common approach in current analysis textbooks. Reading about it in Euler's own words gives context to the exercises and some appreciation of his dexterity with infinitesimals and series, as well as the close connection with $$e$$ as a logarithm base to motivate the definition. Moreover, his series development of $$e^{z}$$ is an interesting alternative to the Taylor series approach students have seen in introductory calculus courses.

The complete project Euler's Rediscovery of $$e$$ (pdf file) is ready for student use, and the LaTeX source code is available from the author by request. A set of instructor notes that explain the purpose of the project and guide the instructor through the goals of each of the individual sections is appended at the end of the student project.

This project is the fifth in A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources that is planned for publication in Convergence, for use in courses ranging from first year calculus to analysis, number theory to topology, and more. Links to other mini-PSPs in the series appear below. The full TRIUMPHS collection includes eleven PSPs for use in a real analysis course.

Acknowledgments

The development of the student project Euler's Rediscovery of $$e$$ has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project with funding from the National Science Foundation’s Improving Undergraduate STEM Education Program under Grants No. 1523494, 1523561, 1523747, 1523753, 1523898, 1524065, and 1524098. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily reflect the views of the National Science Foundation.

References

[1] Euler, L. 1748, Introductio in Analysin Infinitorum, St. Petersburg; English translation by J. Blanton.

# How to Calculate $$\pi$$: Machin's Inverse Tangents, A Mini-Primary Source Project for Calculus II Students

Author(s):
Dominic Klyve (Central Washington University)

Almost every mathematical culture through history seems to have proved, trusted, or suspected that the area of a circle is a fixed constant times the square of its radius. It is maybe not surprising, then, that the last two millennia have seen a seemingly endless array of attempts to calculate this constant (today usually called $$\pi$$) with increasing precision. Despite this history, actually calculating $$\pi$$ is not a significant part of the standard undergraduate curriculum. (Try asking your senior mathematics majors to calculate $$\pi$$ by hand from first principles, even to two decimal places. My own students are not always up to the challenge.)

Calculation of $$\pi$$ - Historical Record Holders [1]

Name Date Number of Digits
Archimedes 250 BCE 2
Ptolemy 150 3
Liu Hui 263 5
Zu Chongzhi 480 7
Jamshīd al-Kāshī 1424 16
Ludoplh van Ceulen 1596 20
Ludoplh van Ceulen 1620 32
Willebrord Snell 1621 35
Christoph Grienberger 1630 38
Abraham Sharp 1699 71
John Machin 1706 100
Thomas Fantet de Lagny 1719 112
Jurij Vega 1789 126

A careful consideration of who has held the record for $$\pi$$ calculation at any time is well beyond the scope of this introduction. I would like, however, to accept as a reasonable proxy the list of calculation records as it appears on Wikipedia on the day I write this. (The skeptical reader may like to know that the list is largely consistent with the list on MacTutor.)

It seems that people made progress fairly steadily for 2000 years, as incrementally more effort was spent and better notation was developed. Then, around 1700, progress in digit calculation accelerated noticeably.

How can we explain this noticeable increase in humanity’s ability to calculate $$\pi$$? It was not the result of simply spending more time and effort on the problem; rather the improvement was a result of a new technology – an understanding of the infinite series of arctangent! One of the first people to put this technology to work was John Machin, a long-time secretary of the British Royal Society.

Machin’s work is fascinating from a modern point of view, as it serves as a bridge between earlier geometric methods of calculating $$\pi$$ and the analytic methods based on infinite series that would dominate in the following centuries.

It seems that no record of this work written by Machin himself has survived, but we do have an account of it written by the English judge Francis Maseres, also a Fellow of the Royal Society [2]. As Maseres recounted the method, Machin realized he could use the formula for the tangent (or arctangent) of the difference of two angles to estimate a given fraction of $$\pi$$, and thus of $$\pi$$ itself. In order to do this, Machin needed to find an angle with two important properties:

• The tangent of the angle is a simple, easy-to-use small fraction; and

• Using the double-angle formula, one can use this angle to find the tangent of an angle very close to $$\pi$$/4.

As a bonus, it would be nice if plugging the fraction into Leibniz's series led to an easy-to-calculate value.

As some readers of Convergence know, Machin succeeded handsomely in this endeavor. Noting the convenient fact that, in Maseres' words, “the tangent of 1/4 of 45° is nearly = 1/5”, Machin not only knew that

$$\pi/4 ≈ 4 * \arctan(1/5)$$,

but was able to use the difference-of-angles formula to find

$$\pi/4 = 4 * \arctan(1/5)\, – \arctan(1/239)$$.

A good picture can reveal an entire argument. Segment AB = arctan(1/5), segment AC = 2arctan(1/5), and segment AD = 4arctan(1/5). Angle AMK = $$\pi$$/4, giving us immediately (if we trust the diagram) that $$\pi/4 ≈ 4 * \arctan(1/5)$$.

In this Primary Source Project, students use Machin’s ideas about difference-of-tangent formulas and rapidly-converging series to gain an understanding of how $$\pi$$ could be calculated. They also gain an appreciation of the relationship between geometric and analytic trigonometry, and can observe directly and numerically how quickly an infinite series (in this case, the Leibniz series for arctangent) converges.

The complete project How to Calculate π: Machin's Inverse Tangents (pdf file) is ready for student use, and the LaTeX source code is available from the author by request. A set of instructor notes that explain the purpose of the project and guide the instructor through the goals of each of the individual sections is appended at the end of the student project.

This project is the sixth in A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources that is planned for publication in Convergence, for use in courses ranging from first year calculus to analysis, number theory to topology, and more. Links to other mini-PSPs in the series appear below. An additional three mini-PSPs on methods for calculuating $$\pi$$, based on works by Archimedes, Leonhard Euler, and the Compte de Buffon, are also planned for inclusion in the full TRIUMPHS collection

Acknowledgments

The development of the student project How to Calculate π: Machin's Inverse Tangents has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project under Grants No. 1523494, 1523561, 1523747, 1523753, 1523898, 1524065, and 1524098. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily reflect the views of the National Science Foundation.

References

[1] Chronology of computation of $$\pi$$. https://en.wikipedia.org/wiki/Chronology_of_computation_of_%CF%80. Accessed August 3, 2018.

[2] Francis Maseres. Scriptores logarithmici; or a Collection of Several Curious Tracts on the Nature and Construction of Logarithms. London, 1791 – 1807.

# Henri Lebesgue and the Development of the Integral Concept: A Mini-Primary Source Project for Undergraduate Analysis Students

Author(s):
Janet Heine Barnett (Colorado State University - Pueblo)

What classes of functions are integrable? The full answer to this question – that it depends on the type of integration used – is an intriguing fact not generally known to our introductory analysis students.

In the earliest days of calculus, the process of integration was considered to be identical with that of finding an antiderivative. While this meant that any expression that was not itself a derivative could not be integrated, this limitation was neither a theoretical or practical concern at that time. Today, we know that antidifferentiable functions necessarily possess the Intermediate Value Property. This result, known as Darboux's Theorem, rules out integrability for basic step functions under the "integration as antidifferentiation" conception. On the other hand, these discontinuous functions are quite easily integrated under the geometric view of "area under the curve" that emerged as the primary conception of the integral over the course of the 18th century.

Lebesgue's diagram for the Riemann integral.

It was this geometric view of an integral that Cauchy sought to capture by defining the integral as the limit of what is today called a Riemann sum, in tribute to Riemann's thorough investigation of conditions for integrability under that definition. But as Lebesgue noted in the introduction to his doctoral dissertation [1]:

It is known that there are derivatives that are not integrable, if one accepts Riemann's definition of the integral; the kind of integration as defined by Riemann does not allow in all cases to solve the fundamental problem of calculus:

Find a function with a given derivative.

It thus seems natural to search for a definition of the integral which makes integration the inverse operation of differentiation in as large a range as possible.

The outcome of Lebesgue's search for such a definition was the integral named in his honor.

The mini-Primary Source Project (PSP) Henri Lebesgue and the Development of the Integral Concept uses excerpts from a relatively non-technical 1927 paper by Lebesgue [2] as a means to consolidate students' understanding of the Riemann integral and its relative strengths and weaknesses. Following a brief overview of the evolution of the integral concept, Lebesgue contrasted the conceptual notions behind the Riemann and Lebesgue integrals. Lebesgue's clever metaphorical comparisons of Riemann's approach to that of an unsystematic merchant "who counts coins and bills at random in the order in which they came to hand" and his own approach to that of "a methodical merchant" who stacks coins of the same denomination together before counting them allowed him to quite naturally bring in the concept of set measure and its role in defining his integral.

Diagram from Lebesgue showing the set $$E_i= \{ x \, \vert \, y_i\le f(x) \le y_{i+1}\}$$.

In keeping with the content objectives of a typical undergraduate introductory analysis course, the PSP's primary content focus is on the definition and properties of the Riemann integral. Reading about Lebesgue's motivations for developing a different type of integration in his own words also allows students to witness the ways in which mathematicians hone various tools of their trade (e.g., definitions, theorems). The project further touches on issues related to the tensions between "logical rigor" and "geometrical intuition" as guiding principles in mathematics. In fact, Lebesgue explicitly described his new definition of the integral as an effort to reconcile these two desirable but conflicting aspects of mathematics. Additionally, this project offers undergraduates a first (if brief) glimpse of the integral that is the current standard in graduate courses and mathematical research --- at least for the time being!

The complete project Henri Lebesgue and the Development of the Integral Concept (pdf) is ready for student use, and the LaTeX source code is available from the author by request. A set of instructor notes that explains the purpose of the project and offers guidance on its implementation is appended at the end of the student project.

This project is the seventh in A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources that is planned for publication in Convergence, for use in courses ranging from first year calculus to analysis, number theory to topology, and more. Links to other mini-PSPs in the series appear below. The full TRIUMPHS collection includes eleven PSPs for use in a real analysis course.

Acknowledgments

The development of the student project Henri Lebesgue and the Development of the Integral Concept has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project with funding from the National Science Foundation's Improving Undergraduate STEM Education Program under grant number 1523494. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily represent the views of the National Science Foundation. The author also wishes to thank George W. Heine III for recreating the diagrams used in this article.

References

[1] Lebesgue, H. Intégrale, Longueur, Aire. PhD thesis, Université de Paris, Milan: Bernandon de C. Rebeschini. 1902.

[2] Lebesgue, H. Sur le développement de la notion d’intégrale. Revue de Metaphysique et de Morale, 34 (2):149-167. English translation by Kenneth O. May, Classics of Mathematics (ed. R. Calinger), Prentice-Hall, New Jersey, 1995, pp. 762-765. 1927.

# Seeing and Understanding Data: A Mini-Primary Source Project for Students of Statistics

Author(s):
Charlotte Bolch (University of Florida) and Beverly Wood (Embry-Riddle Aeronautical University)

Mathematicians and scientists included drawings in their work before the first statistical graphs were invented. However, their illustrations were meant to depict quantifiable relationships rather than exposing statistical variability. The uses of pictorial representations or charts to convey trends among variable measurements dates back to at least the late 10th century. However, this means of communication did not gain the widespread appreciation it enjoys today until technological advances in the 21st century stimulated the growth of a new field, data visualization. The mini-Primary Source Project (PSP) Seeing and Understanding Data provides students the opportunity to explore the evolution of statistical graphs and visual displays across time and to think critically about how data are displayed and interpreted.

The mini-PSP begins by introducing students to two graphical displays, one from circa 1000 CE (Figure 1) and the other from the mid-15th century (Figure 2), that have features similar to graphs with which they are more familiar. Then, students explore the origins of some now-common statistical graphs in the work of William Playfair (1759-1823), who was apparently the first to publish examples of the pie chart, bar chart, and statistical line chart [4]. Although Playfair was well-acquainted with major figures of the Scottish Enlightenment, a career checkered by often-illegal activities meant that he and his innovations in communicating data to a general audience remained largely obscure in Great Britain and only slightly better-known in France. It was only in the late 19th century that these graphical methods began to be more widely used by statisticians. (See [1] for a reprint of Playfair’s Commercial and Political Atlas.)

In the 19th century, the professional discipline of statistics and visual representations of data often developed in concert. Graphs came to be used not only to illustrate trends or experiments in science, mathematics, economics, and industry, but also to enhance arguments for social reforms. The mini-PSP explores one such instance, the hand-colored displays Florence Nightingale (1820-1910) prepared for her pivotal report to Britain’s Minister of War to show the proportion of deaths from preventable disease during the Crimean War [3]. Her dramatic data displays effected changes to all British Army hospitals–changes also incorporated into the U.S. Army hospitals during the American Civil War. Students investigate both the story and the technical aspects of production for this influential visualization, as well as Charles Minard’s equally elegant display of data from Napoleon’s ill-conceived march on Moscow.

The mini-PSP closes with a look at the production of data displays supported by modern technology. Students then use the nine characteristics of graphical excellence espoused by Edward Tufte (1942– ) to re-examine the primary source visualizations in the project, including the graphs shown in Figures 1 and 2 above and five other seminal graphs:

• Exports and Imports of Scotland to and from different parts for one Year from Christmas 1780 to Christmas 1781, Bar Graph by William Playfair

• Statistical Representation of the United States of America, Pie Chart by William Playfair

• Chart Shewing the Extent, Population & Revenues, of the Principal Nations of Europe in the Order of their Magnitude, Circle Graph by William Playfair

• Causes of mortality in the Army in the East, Coxcomb Diagram by Florence Nightingale

• Successive losses of men in the French Army during the Russian Campaign of 1812–1813, Figurative Map by Charles Joseph Minard

The final project task requires access to the internet (in or out of the classroom) for students to investigate Gapminder’s interactive display of income versus life expectancy data for the whole world over 200 years.

As Hans Rosling once said, “Most of us need to listen to the music to understand how beautiful it is. But often that’s how we present statistics: we just show the notes, we don’t play the music” [2]. Through this mini-PSP, we hope students see and understand how statistical graphs and data visualizations provide the essential music for how statistics is presented and interpreted by the world. Instructors of introductory statistics and data science courses at the high-school and college levels can implement this project over two-three class periods using a combination of small-group work and whole-class discussion. Some project tasks, especially those requiring use of the internet, also work well as individually-assigned out-of-class work. This mini-PSP is also suitable for mathematics education students, and can be completed by middle school students with more assistance and direction from the instructor.

The complete project Seeing and Understanding Data (pdf) is ready for student use and the LaTeX source code is available from the authors by request. Instructor notes are provided to explain the purpose of the project and guide the instructor through each of the individual sections of the project.

This project is the eighth in A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources that is planned for publication in Convergence, for use in courses ranging from first year calculus to analysis, number theory to topology, and more. Links to other mini-PSPs in the series appear below. The full TRIUMPHS collection also includes one other PSP for use in teaching statistics, the full-length project entitled Quantifying Certainty: The p-value.

Acknowledgements

The development of the student project Seeing and Understanding Data has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project with funding from the National Science Foundation’s Improving Undergraduate STEM Education Program under Grants No. 1523494, 1523561, 1523747, 1523753, 1523898, 1524065, and 1524098. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily reflect the views of the National Science Foundation.

References

[1] Playfair, William (2005). The Commercial and Political Atlas and Statistical Breviary. Edited and introduced by Howard Wainer and Ian Spence. New York: Cambridge University Press.

[2] Rosling, Hans (2007, June). Turning Statistics into Knowledge. Paper presented at the Second OECD World Forum on “Statistics, Knowledge and Policy”, Istanbul, Turkey.

[3] Stone, M. (2001). Florence Nightingale. In Heyde, C.C. & Seneta, E. (Eds.), Statisticians of the Centuries (171-175). New York, NY: Springer-Verlag New York, Inc.

[4] Spence, I. & Wainer, H. (2001). William Playfair. In Heyde, C.C. & Seneta, E. (Eds.), Statisticians of the Centuries (105-110). New York, NY: Springer-Verlag New York, Inc.

[5] Unknown (1010). De cursu per zodiacum. https://commons.wikimedia.org/wiki/File:Clm_14436_ecliptic_diagram.png#/. (Online; accessed November 2, 2018).

[6] van Langren, Michaël Florent. (1643). Graph of statistical data, showing the wide range of estimates of the distance in longitude between toledo and rome, 1643.  https://commons.wikimedia.org/wiki/File:Grados_de_la_Longitud.jpg. (Online; accessed November 2, 2018).

# The Origin of the Prime Number Theorem: A Primary Source Project for Number Theory Students

Author(s):
Dominic Klyve (Central Washington University)

Near the end of the eighteenth century, Adrien-Marie Legendre (1752–1833) and Carl Gauss (1777–1855) seemingly independently began a study of the primes–more specifically, of what we now call their density.  It would seem fairly clear to anyone who considered the matter that prime numbers are more rare among larger values than among smaller ones, but describing this difference mathematically seems not to have occurred to anyone earlier. Indeed, there's arguably no a priori reason to assume that there is a nice function that describes the density of primes at all. The mini-Primary Source Project (PSP) The Origin of the Prime Number Theorem provides students with an introduction to this problem through the writing of Gauss and Legendre.

Late in his life (Christmas Day, 1849), Gauss wrote a letter to his colleague Johann Encke about prime numbers. He reported that he had been interested in primes when he was young, and that in fact he had both tabulated the number of primes in various intervals himself, and then spent some time studying a table of primes that had recently been compiled by John Lambert. As early as 1792 or 1793, Gauss claimed, he had conjectured that the number of primes below a bound $$n$$ was, in his notation, $$\int \frac{dn}{\log n}$$.

Today we know that Gauss was correct, but we write his conjecture differently. Using more modern notation, and letting as usual $$\pi(x)$$ denote the number of primes less than or equal to $$x$$, we have
$\pi(x) \sim \int_2^x \frac{dt}{\log t} \sim \frac{x}{\log x}.$

(The function $$\mbox{li}(x) = \int_2^x \frac{dt}{\log t}$$ is often referred to as the logarithmic integral. The notation $$f(x) \sim g(x)$$ indicates that  $$\lim_{n \to \infty} \frac{f(x)}{g(x)} = 1$$.)

Gauss told no one at the time that he was thinking about prime numbers, and thus Legendre, in the second edition of his Essai sur la Théorie des Nombres (Essay on Number Theory) [1], had good reason to suspect he was the first person to consider the question of the number of primes below a given bound. He began by  noting one of the greatest puzzles in modern number theory–the sequence of primes is indeed "extremely irregular," and seems almost random on a small scale, but on a larger scale they seem to follow a rather predictable pattern.  Here is how Legendre described  this pattern (the English translation is mine):

On a very remarkable law observed in the enumeration of prime numbers

Although the sequence of prime numbers is extremely irregular, one can however find, with a very satisfying precision, how many of these numbers there are from $$1$$ up to a given limit $$x$$. The formula that resolves this question is
$y = \frac{x}{\log(x) - 1.08366} \ldots.$

Legendre then presented a table comparing the values predicted by his formula with the true count, as determined by a recently-published table of primes (Juri Vega's Thesaurus logarithmorum [2]).

Gauss claimed merely to have looked at the data and seen the pattern; his complete statement reads "I soon recognized that behind all of its fluctuations, this frequency is on the average inversely proportional to the logarithm." Legendre gave even less indication of the origin of his estimate; in particular, his rather precise constant of $$1.08366$$ has, as far as I know, never been satisfactorily explained. In the mini-PSP The Origin of the Prime Number Theorem, students explore how Legendre and Gauss may have arrived at their conjectures, compare the similar (though not identical) estimates for the number of primes up to $$x$$ given by each, and examine some of the ideas related to different formulations of the Prime Number Theorem.  Using the letter written by Gauss, they then examine the error in their respective estimates.

The complete project The Origin of the Prime Number Theorem (pdf) is ready for student use and the $$LaTeX$$ source code is available from the author by request. Instructor notes are provided to explain the purpose of the project and guide the instructor through implementation of the project.

This project is the ninth in A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources that is planned for publication in Convergence, for use in courses ranging from first year calculus to analysis, number theory to topology, and more. Links to other mini-PSPs in the series appear below, including two additional mini-PSPs in number theory. The full TRIUMPHS collection also offers five more extensive “full-length” PSPs for use in teaching number theory.

Acknowledgments

The development of the student project Seeing and Understanding Data has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project with funding from the National Science Foundation’s Improving Undergraduate STEM Education Program under Grants No. 1523494, 1523561, 1523747, 1523753, 1523898, 1524065, and 1524098. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily reflect the views of the National Science Foundation.

References

[1] Adrien-Marie Legendre, Essai sur la théorie des nombres (seconde edition), Courcier: Paris, 1808.
[2] George Vega and Adriaan Vlack,
Thesaurus logarithmorum completus: ex Arithmetica Logarithmeticam, et ex Trigonometria Artificiali Adriani Vlacci collectus, etc. / Vollständige Sammlung Grösserer Logarithmisch-trigonometrischer Tafeln nach Adrian Vlack’s Arithmetica Logarithmetica und Trigonometria Artificialis (Complete collection of logarithmic-trigonometric tables, from the Arithmetica Logarithmica and Trigonometria Artificialis of Adrian Vlack, etc.), Weidmannschen Buchhandlung,1794.

# The Cantor Set Before Cantor: A Mini-Primary Source Project for Analysis and Topology Students

Author(s):
Nicholas A. Scoville (Ursinus College)

Historians and others have long suspected that if a mathematical object or theorem is named after a person, then this is all but a guarantee that the object or theorem was not originally due to this person. The study of Pell’s equation goes back about a thousand years before Pell was born. L’Hôpital infamously bought the right to the discovery of the rule now bearing his name from Johann Bernoulli. Burnside’s Lemma is so notorious for having been known before Burnside that it has come to be called "The Lemma that is not Burnside's."

 Henry Smith (1826–1883) The Cantor set, named after Georg Cantor (1845–1918), is no exception to this rule. While Cantor defined the set now bearing his name in an 1883 paper, a version of it was defined several years earlier in 1875 by Henry John Stephen Smith in a paper that appeared in the Proceedings of the London Mathematical Society entitled “On the Integration of Discontinuous Functions” [1]. Smith (1826–1883) was an Irish mathematician and a professor at Oxford. He made significant contributions to number theory as well as analysis, and is best known today for the Smith normal form of a matrix. In his 1875 paper, Smith took up the question of when a discontinuous function can be integrable. While it is easy to find examples of integrable functions which are discontinuous at finitely many points, it is more difficult to construct integrable functions which are discontinuous at infinitely many points. Smith developed a method of constructing such a function. He proved that any function which coincides with an integrable function everywhere other than a nowhere dense set is integrable. He then gave several examples, including a version of the Cantor set, of nowhere dense sets. Applying his theorem, these sets provide examples of integrable functions which are discontinuous on infinitely many points.

The mini-Primary Source Project (mini-PSP) The Cantor Set Before Cantor begins with the question of how discontinuous a function can be while still being integrable, and ends with the construction of a version of the generalized Cantor set. In the middle of the project, students see Smith’s theorem concerning the construction of integrable functions with discontinuities at every point of a nowhere dense set. After working through some initial definitions and exercises, students are exposed to three of Smith’s examples of nowhere dense sets. The project has the students show that these sets are nowhere dense, and then use Smith’s theorem to construct a function which is discontinuous on that set yet still integrable. Smith’s third example is his version of the Cantor set, a set which illustrates that our initial intuition about how objects “ought” to behave is not always so close to the reality of things. While topologically interesting in its own right, the Cantor set also serves the special purpose of providing an example of an extremely “large” set on which a function can be discontinuous and integrable. In this sense, the project acts as a bridge between analysis and topology that can help students see the connections between the disciplines.

The Cantor set after seven iterations

The complete project The Cantor Set Before Cantor (pdf) is ready for student use and the LaTeX source code is available from the author by request. Instructor notes are provided to explain the purpose of the project and guide the instructor through its implementation.

This project is the tenth in A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources that is planned for publication in Convergence, for use in courses ranging from first year calculus to analysis, number theory to topology, and more. Links to other mini-PSPs in the series appear below, including the topology mini-PSP Connecting Connectedness. The full TRIUMPHS collection also offers four other mini-PSPs and two more extensive “full-length” PSPs for use in teaching topology.

Acknowledgments

The development of the student project The Cantor Set Before Cantor has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project with funding from the National Science Foundation’s Improving Undergraduate STEM Education Program under Grants No. 1523494, 1523561, 1523747, 1523753, 1523898, 1524065, and 1524098. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily reflect the views of the National Science Foundation.

References

[1] H. J. S. Smith. On the Integration of Discontinuous Functions. Proc. Lond. Math. Soc., 6:140-153, 1874/75.

# Euler’s Calculation of the Sum of the Reciprocals of the Squares: A Mini-Primary Source Project for Calculus II Students

Author(s):
Kenneth M Monks (Front Range Community College–Boulder County Campus)

A central theme of most second-semester calculus courses is that of infinite series. Students typically learn to classify infinite series as convergent or divergent via a lengthy list of convergence tests. In some cases, they can proceed to evaluate a convergent series by strategically plugging an $$x$$ value from the relevant interval of convergence into a power series formula. What is often missing is any indication that it is possible to evaluate an infinite series in any other manner!

Fortunately, in his 1740 paper De Summis Serierum Reciprocarum, Leonhard Euler (1707–1783) provided an example of a nontrivial evaluation of an infinite series that is beautiful and does not require significant extension of the topics one would normally cover in an introductory treatment of power series. Known as The Basel Problem, the evaluation of the series $\sum_{n=1}^\infty \frac{1}{n^2}=1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\cdots$ proved to be quite a challenge. For example, Jacob Bernoulli (1655–1705) was able to prove the series converged to a number less than $$2$$, but the exact value eluded him [Bernoulli 1713].

In this project, we guide students through Euler's incredibly clever proof that this series converges to $$\pi^2/6$$. Although issues related to series convergence were viewed differently in the 18th century, today's standard series convergence tests are used heavily throughout the project. No prerequisite knowledge is required to understand the proof itself beyond the power series for sine and basics from precalculus such as finding zeros of functions and factoring polynomials. While this project is intended for an introductory calculus course, it also makes an ideal starting point for a discussion of the Riemann zeta function or the Weierstrass Factorization Theorem in a complex analysis course, or a discussion of generating functions in a combinatorics course.

The complete project Euler's Calculation of the Sum of the Reciprocals of the Squares (pdf) is ready for student use, and the LaTeX source code is available from the author by request. A set of instructor notes that explain the purpose of the project and guide the instructor through the goals of each of the individual sections is appended at the end of the student project.

This project is the eleventh in A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources that is planned for publication in Convergence, for use in courses ranging from first year calculus to analysis, number theory to topology, and more. Links to other mini-PSPs in the series appear below, including two additional mini-PSPs for first-year calculus. The full TRIUMPHS collection also offers six other mini-PSPs and one more extensive full-length PSP for use with students of calculus.

Acknowledgments

The development of the student project Euler's Calculation of the Sum of the Reciprocals of the Squares has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project with funding from the National Science Foundation’s Improving Undergraduate STEM Education Program under Grants No. 1523494, 1523561, 1523747, 1523753, 1523898, 1524065, and 1524098. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily reflect the views of the National Science Foundation.

References

Bernoulli, J. (1713). Ars conjectandi, opus posthumum: accedit Tractatus de seriebus infinitis, et
epistola Gallice scripta de ludo pil reticularis
.
Basile: Impensis Thurnisiorum.

Euler, L. (1740). De summis serierum reciprocarum. Commentarii academiae scientiarum Petropolitanae, 7:123134.

# Completing the Square: From the Roots of Algebra, A Mini-Primary Source Project for Students of Algebra and Their Teachers

Author(s):
Daniel E. Otero (Xavier University)

Nearly every student of mathematics, after receiving years of training in the rules and procedures of arithmetic, enters the realm of "higher" mathematics through the study of algebraic problem solving, the finding of unknown quantities from known arithmetical conditions on those quantities. Algebra is a staple of the secondary school curriculum around the world, and a standard rite of passage for students in this curriculum is some form of mastery of the process of factoring polynomials, especially quadratic polynomials. Learning to factor quadratics is a precursor to a complete treatment of quadratic equations and their solutions, including the procedure known as "completing the square" and culminating with the well-known quadratic formula: given a quadratic equation of the form $$ax^2+bx+c=0$$ (with $$a\neq0$$), its solutions are given by $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$ The mini-Primary Source Project (mini-PSP) Completing the Square: From the Roots of Algebra presented here is designed to give students a deep understanding of the method of completing the square, which serves as a bridge between the method of factoring and the quadratic formula.

Beginning algebra students naturally focus their attention on mastering the procedures of equation solving and learning to get the correct answers, ignoring questions like "Why and how do the procedures work?" For them, completing the square can involve procedural steps that mysteriously produce the required sought-for answers, and the quadratic formula can act like a runic talisman that magically generates the right numbers that solve the given equation. And even if the question "Why?" is seriously considered in this context, many textbooks answer with a symbolic derivation that is complicated and unsatisfying. The best answer to this question is quite naturally found in the history of the development of this method.

Although problems of quadratic type have been posed and solved for thousands of years, the systematic approach to algebraic problem-solving goes back to the "Father of Algebra," Muḥammad ibn Mūsā al-Khwārizmī (ca 780–850 CE), a ninth-century scholar who wrote in Arabic in the then-young city of Baghdad under the patronage of one of the great caliphs of the Islamic Abbasid Empire. Written in about the year 825, al-Khwārizmī 's extremely influential work on the subject, with the title al-Kitāb al-mukhtaar fī hisāb al-jabr wal-muqābala (The Compendious Book on Calculation by Restoration and Reduction), better known today simply as Algebra, instructs his readers how to find the roots of an equation. But al-Khwārizmī's equations are ones without symbols; they are expressed entirely in words. This rhetorical algebra of al-Khwārizmī provides a careful description of the method we call completing the square, along with a clear geometric demonstration of how the method works that involves completing a real (geometric) square!

A page from an Arabic copy of al-Khwārizmī's Algebra,

In the project Completing the Square: From the Roots of Algebra, students work through selections from al-Khwārizmī's Algebra, using text from two English translations of the work [1,3]. In a pair of appendices, students can then further explore these ideas through (1) a derivation of the quadratic formula, and/or (2) consideration of when a quadratic equation produces complex roots. The project is meant to serve multiple needs: it can be used by students who are learning algebraic methods for solving quadratic equations for the first time; by future high school mathematics teachers who will be responsible for teaching algebra in their own classrooms; and by students in a general history of mathematics course as an introduction to the role of early Islamic-era mathematics in the development of algebra.

The complete project Completing the Square: From the Roots of Algebra (pdf is ready for student use, and the LaTeX source code is available from the author by request. A set of instructor notes that explain the purpose of the project and guide the instructor through the goals of each of the individual sections is appended at the end of the student project.

This project is the twelfth in A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources that is planned for publication in Convergence, for use in courses ranging from first year calculus to analysis, number theory to topology, and more. Links to other mini-PSPs in the series appear below. The full TRIUMPHS collection also offers dozens of other mini-PSPs and a similar number of more extensive full-length PSPs which are meant for other topics across the undergraduate mathematics curriculum, including a longer version of this project (entitled Solving Equations and Completing the Square: From the Roots of Algebra) that includes an introduction to the rhetorical algebra of al-Khwārizmī and a deeper exploration of his solutions to quadratic equations.

### References

[1] Muḥammad ibn Mūsā al-Khwārizmī. The Algebra of Moḥammed ben Mūsā, Translated and Edited by Frederic Rosen. Oriental Translation Fund, London, 1831.

[2] Victor Katz, Annette Imhausen, Eleanor Robson, Joseph W. Dauben, Kim Plofker, and J. Lennart Berggren, editors. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press, Princeton, NJ, 2007.

[3] Rushdī Rāshid. Al-Khwārizmī: The Beginnings of Algebra. Saqi, London, 2009.

### Acknowledgments

The development of the student project Completing the Square: From the Roots of Algebra has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project with funding from the National Science Foundation’s Improving Undergraduate STEM Education Program under Grants No. 1523494, 1523561, 1523747, 1523753, 1523898, 1524065, and 1524098. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily reflect the views of the National Science Foundation.

# Regression to the Mean: A Mini-Primary Source Project for Statistics Students

Author(s):
Dominic Klyve (Central Washington University)

One of the greatest fears in sports performance is that, following a very successful run (of games, shots, field goal kicks, etc.), an individual or team may be “jinxed” and will revert to more average performance. In baseball this has happened so often to second-year players that there is a name for it: the feared “sophomore slump.”

On a seemingly unrelated note ...

Oft-repeated in history books is the lamentation that the progeny of a great man (it does usually seem to be a man in these books) failed to live up to the talent / hard work / brilliance of their parent. The great founding king gives way to a middling prince. Or, in a less regal example, American historian Paul Nagel has described how four generations of the Adams family betrayed an inexorable “Descent from Glory” [Nagel].

Each of these phenomena (from quite unrelated fields) can be explained by the same underlying principle, which is today known as regression to the mean. It states that, given a repeated set of observations, very extreme values will be followed by less extreme ones. This tendency was first noted by Englishman Francis Galton in 1886. Galton seems not only to have been the first person to pose the question of why regression occurs, but in the same paper, he became the first to give an answer. The mini-Primary Source Project presented here offers students the opportunity to learn about regression to the mean by reading from his pioneering paper on the topic.

 Francis Galton (1822–1911) himself is remembered for many things, but for our purposes it may be best to think of him as one of history’s all-time-champion measurers. Nothing that could be quantified escaped his interest, and he was unafraid to use his measurements to draw sweeping conclusions about the world. He studied the lifespan of monarchs and concluded, on the assumption that more people pray for them than pray for average people, that prayer is ineffective in prolonging life. He collected weather data and constructed the first weather map, discovering the phenomenon of “anticyclones” in the process. He even created a “beauty map” of Britain by walking through different cities and making secret records of the attractiveness of women he passed using a device that he called, seemingly with no sense of irony, a “pricker” [Holt]. With an inheritance left to him by his father, Galton gave full rein to his desire to measure things, setting up an “anthropometric laboratory” in London in 1884. He advertised widely, and “thousands of people streamed in and obligingly submitted to measurement of their height, weight, reaction time, pulling strength, color perception, and so on” [Holt, p. 57]. Sir Francis Galton, 1890s Photograph by Eveleen Myers (née Tennant), Public Domain
 Galton's 1861 Weather Map Public Domain In our age of “big data,” many people believe that the company or individual with the best data will have a competitive advantage. More than a century ago, Francis Galton wanted to take many measurements because he believed they would tell him something interesting. Happily, he was correct; with access to an unprecedented amount of anthropometric data, he began to draw conclusions that would otherwise have been impossible, and he thereby became the first person to note the phenomenon of regression. In the Primary Source Project Regression to the Mean, students read parts of Galton’s paper, “Regression Towards Mediocrity in Hereditary Stature” [Galton], to explore the phenomenon and the cause of regression. They are given the opportunity to reason from Galton’s original data of parental and offspring heights, and to explore probabilistic thinking in this setting. Via an optional section of the project, students can also explore errors in reasoning that arise in our world today from a lack of understanding of regression. The complete project Regression to the Mean (pdf file) is ready for student use, and the LaTeX source code is available from the author by request. A set of instructor notes that explain the purpose of the project and guide the instructor through the goals of each of the individual sections is appended at the end of the student project.

This project is the thirteenth in A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources that is planned for publication in Convergence, for use in courses ranging from first-year calculus to analysis, number theory to topology, and more. Links to other mini-PSPs in the series appear below, including the statistics mini-PSP Seeing and Understanding Data. The full TRIUMPHS collection also offers a more extensive “full-length” PSP for use in teaching the p-value.

Acknowledgments

The development of the student project Euler's Calculation of the Sum of the Reciprocals of the Squares has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project with funding from the National Science Foundation’s Improving Undergraduate STEM Education Program under Grants No. 1523494, 1523561, 1523747, 1523753, 1523898, 1524065, and 1524098. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily reflect the views of the National Science Foundation.

References

Galton, Francis. "Regression towards mediocrity in hereditary stature." The Journal of the Anthropological Institute of Great Britain and Ireland 15 (1886): 246–263.

Holt, Jim. “Sir Francis Galton, the Father of Statistics … and Eugenics." In When Einstein Walked with Gödel: Excursions to the Edge of Thought, pp. 51–68. Farrar, Straus and Giroux, 2018.

Nagel, Paul C. Descent from Glory: Four generations of the John Adams family. Harvard University Press, 1999.

# Investigations Into d'Alembert's Definition of Limit: A Mini-Primary Source Project for Students of Real Analysis and Calculus 2

Author(s):
David Ruch (Metropolitan State University of Denver)

Students of mathematics struggle mightily with the meaning and definition of limit, whether in introductory courses on calculus or in real analysis. This seems fair, since mathematicians themselves have not always agreed on the subtleties of the concept, nor on the need to capture that concept in the form of a precise algebraic definition such as the one that we use today. A major problem in the  development of today's definition was finding a useful bridge between the infinitesimal and finite worlds: how to translate ideas like "approach a limiting value" and "as close as we please" into precise algebraic inequalities with clear logical quantifiers. This mini-Primary Source Project (mini-PSP) explores the limit definition given and used by Jean-Baptiste le Rond d'Alembert (1717–1783).

 Portrait of d'Alembert Convergence Portrait Gallery Early chapters of d'Alembert's biography read like something out of Masterpiece Theater. He was born out of wedlock and left as an infant at the church Saint Jean le Rond in Paris. His salonniére mother Claudine Guérin de Tencin never acknowledged him, and his father Louis-Camus Destouches found another woman to raise young Jean. Destouches died in 1726, but left funds for Jean's education. D'Alembert did well in school and became active as an adult in the philosophy, literature, science and mathematics of his day, standing "at the very heart of the Enlightenment with interests and activities that touched on every one of its aspects" [Hankins]. He co-edited the quintessential work of the French Enlightenment, Diderot’s famous Encyclopédie, for a time, and wrote a number of its articles. Excerpts from two of these, “Limite” and “Calcul différential,” are used in the student project.
 There are two versions of this mini-PSP: one for students of real analysis, and a more mild version for use in a first-year calculus course. D'Alembert's definition of limit has pedagogical value for modern students in both courses because it is reasonably close to a verbal version of our modern definition, but with two striking differences. In his 1754 article on the limit [D'Alembert 1754a], d'Alembert asserted (boldface added): One says that a magnitude is the limit of another magnitude, when the second may approach the first more closely than by a given quantity, as small as one wishes, moreover without the magnitude approaching, being allowed ever to surpass the magnitude that it approaches;           ...... Strictly speaking, the limit never coincides, or is never equal to the quantity of which it is the limit; but the latter approaches it ever more closely, and may differ from it by as little as one wishes. The circle, for example, is the limit of the inscribed and circumscribed polygons; for strictly it never coincides with them, though they may approach it indefinitely. In other words, d'Alembert did not allow the converging quantity to ever reach or oscillate about the limiting value. Students are confronted with these differences in both versions of the project. d’Alembert's original Encyclopédie article, “Limite” gallica.bnf.fr / Bibliothèque nationale de France

Notably, d'Alembert's example in his 1754 article is sequential in nature, examining sequences of inscribed and circumscribed polygons converging to a circle. After an introduction with some historical background, the mini-PSP also focuses on the limit of a sequence. Some popular textbooks for real analysis treat sequential limits first, so the project can be used as an introduction to this core topic in either course.  Proof-related ideas are omitted entirely from the Calculus 2 version of the mini-PSP, which focuses instead on basic examples of sequence convergence. In the real analysis version, students are guided from d'Alembert's verbal description of a sequence limit to a precise modern version using inequalities and quantifiers. That version further investigates two limit properties stated by d'Alembert: uniqueness of limits, and the convergence of a product of two converging quantities. D'Alembert gave a nice proof of the uniqueness result in his “Calcul différential” article, but using phrases such as "approach X as closely as we desire" and "the difference between Y and X can be as small as wished" [D'Alembert 1754b]. Students are asked to translate his theorem statements and give proofs of each in modern language.

Octagon and square inscribed in circle, showing difference in perimeters.
Created by KSmrq [CC BY-SA (http://creativecommons.org/licenses/by-sa/3.0/)]

Both versions of the project Investigations Into d'Alembert's Definition of Limit are ready for student use, respectively subtitled Real Analysis Version (pdf file) and Calculus 2 Version (pdf file). The LaTeX source code of each is available from the author by request. A set of instructor notes offering background and practical advice for the use of the project in the classroom is appended at the end of both student versions of the project.

This project is the fourteenth in A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources that is planned for publication in Convergence, for use in courses ranging from first-year calculus to analysis, number theory to topology, and more. Links to other mini-PSPs in the series appear below. The full TRIUMPHS collection includes twelve PSPs for use in a real analysis course, and seven PSPs for use in a Calculus 2 course.

Acknowledgments

The development of the student project Investigations Into d'Alembert's Definition of Limit has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project with funding from the National Science Foundation’s Improving Undergraduate STEM Education Program under Grants No. 1523494, 1523561, 1523747, 1523753, 1523898, 1524065, and 1524098. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily reflect the views of the National Science Foundation. The author gratefully acknowledges this support, with special thanks to TRIUMPHS PI Janet Heine Barnett, who helped with suggestions for producing the Calculus 2 version of the original project.

References

Jean le Rond d’Alembert. Limite (mathématiques). In Encyclopédie ou Dictionnaire raisonné des sciences, des arts et des métiers, volume 9, page 542. Paris, 1754a. Translation by Jacqueline Stedall, in Mathematics Emerging: A Sourcebook 15401900, pp. 297–298, Oxford: Oxford University Press, 2008.

Jean le Rond d’Alembert. Calcul différentiel. In Encyclopédie ou Dictionnaire raisonné des sciences, des arts et des métiers, volume 9, pages 985–988. Paris, 1754b. Translation by Gregory Bringman, in The Encyclopedia of Diderot & d’Alembert Collaborative Translation Project, Ann Arbor: Michigan Publishing, University of Michigan Library, 2003. https://quod.lib.umich.edu/d/did/did2222.0001.091.

Thomas L. Hankins. Jean d’Alembert: Science and the Enlightenment. New York: Gordon and Breach, 1990.

# Braess’ Paradox in City Planning: A Mini-Primary Source Project for Multivariable Calculus Students

Author(s):
Kenneth M Monks (Front Range Community College – Boulder County Campus)

Optimization, via both the second derivative test and Lagrange multipliers, is a standard topic in undergraduate multivariable calculus courses. There is also a well-known canon of lovely applications of these techniques, including least squares lines of best fit, Cobb-Douglas production functions, and three-dimensional packaging/container problems.

An entirely separate paradigm for optimization—combinatorial optimization—also exists. While it would be excessive to venture heavily into this topic in a multivariable calculus course, it is possible to offer students a small glimpse into the world of graph theory that further allows them to witness a setting in which either discrete or continuous techniques can be used. This easy detour can be accomplished by looking at a fairly recent, but highly accessible, bit of mathematics known as Braess' Paradox. Braess' Paradox is a counterintuitive phenomenon in which the removal of an edge in a congested network strangely results in improved flow. This is often witnessed in traffic flow settings. For example, when New York City's busy 42d Street was closed to automobiles for an Earth Day celebration, it did not generate the gridlock that people expected, but rather resulted in improved flow [Kolata,1990]. In fact, New York City ultimately decided to make this area permanently pedestrian-only.

Opening shot from time-lapse video “Transit Day 1990 Auto-Free 42nd Street," filmed on 22 April 1990.

The primary source project Braess’ Paradox in City Planning: An Application of Multivariable Optimization walks the student through a guided reading of excerpts from the 1968 paper “Uber ein Paradoxon aus der Verkehrsplanung” [Braess, 1968] (or, in English, “On a paradox of traffic planning” [Braess, 2005]) in which this intriguing phenomenon was first studied. Written by Dietrich Braess, currently professor emeritus at Ruhr Universität Bochum, this paper describes a method for detecting this paradox in a network using the framework of combinatorial optimization. When Braess’ paper first appeared, the application of mathematics to traffic planning was still a relatively new idea.1 Today, the paradox that bears his name continues to be studied by mathematical researchers who are interested, for example, in exploring the conditions under which it will not occur. Braess’ paradox is also put to practical use by transportation specialists who are responsible for designing today’s real-life traffic networks. As catalogued by Nagurney & Nagurney [forthcoming], additional applications of Braess’ paradox have been introduced into the modeling of telecommunication networks and the Internet, as well as the study and design of electrical power systems and electronic circuits, mechanical and fluid systems, metabolic networks and ecosystems, and even sports analytics!

Braess' Paradox simulation involving four routes, two of which (in yellow and green) cross the creek bridge.
Observe that the average travel time never exceeds 2 units before the bridge roadblock is removed,
then becomes larger than 2 units once the bridge is opened.
Movie produced by author using the interactive traffic simulator developed by Brian Hayes;

In the student project based on his classic paper, we first retrace Braess' work, and then see how the examples he provided can also be analyzed using standard optimization techniques from a multivariable calculus course. Thus, we provide the student simultaneously with exposure to a second optimization framework and with practice applying the standard multivariable calculus optimization techniques.

The complete project Braess' Paradox in City Planning: An Application of Multivariable Optimization (pdf) is ready for student use, and the LaTeX source code is available from the author by request. A set of instructor notes that explain the purpose of the project and guide the instructor through the goals of each of the individual sections is appended at the end of the student project. This project is the fifteenth in A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources appearing in Convergence, for use in courses ranging from first-year calculus to analysis, number theory to topology, and more. Links to other mini-PSPs in the series appear below, including four additional mini-PSPs on topics from first-year calculus courses. The full TRIUMPHS collection offers a total of eleven mini-PSPs and one more extensive “full-length” PSP for use with students of calculus.

Note

1. The first efforts to formulate traffic flow problems mathematically began only in the 1950s, and two publications from that decade [Wardrop 1952, Beckmann et al. 1956] represented the state of the art when Braess himself entered the field. Surprisingly, Braess was completely unaware of those important prior works! His interest in the mathematical modeling of traffic flow instead was inspired by a 1967 seminar talk in which German mathematician W. Knödel presented a certain algorithm that roused Braess’ curiosity. At the time, Braess was 29 years old and had only recently turned to the study of mathematics after completing his doctorate in theoretical physics just three years earlier. Nearly 40 years later, in 2006, Braess delivered his first North American lecture on the paradox named in his honor at the Virtual Center for Supernetworks at University of Massachusetts Amherst. In that lecture, he remarked that both his lack of knowledge of the then-current state of transportation science and his background in physics, which had trained him to look for a counterintuitive symmetry-breaking argument, were important factors in shaping the work that led to his discovery of the paradox that now bears his name.

Acknowledgments

The author would like to thank his advisor Alexander Hulpke and former graduate school classmate Cayla McBee for initial exposure to this wonderful topic! Thanks also to Brian Hayes for creating the simulator which was used to generate the animation above, and for granting permission for its use to produce that animation and to provide a link to the related article Hayes 2015a.

The development of the student project Braess' Paradox in City Planning: An Application of Multivariable Optimization has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project with funding from the National Science Foundation’s Improving Undergraduate STEM Education Program under Grants No. 1523494, 1523561, 1523747, 1523753, 1523898, 1524065, and 1524098. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily reflect the views of the National Science Foundation. The author gratefully acknowledges this support, with special thanks to TRIUMPHS PI Janet Heine Barnett, who provided assistance with the historical content of the project.

References

Martin J. Beckmann, C. B. McGuire, and C. B. Winsten. 1956. Studies in the Economics of Transportation. New Haven, CT: Yale University Press.

Dietrich Braess. 1968. Uber ein Paradoxon aus der Verkehrsplanung. Unternehmensforschung, 12: 258–268.

Dietrich Braess. 2005. On a paradox of traffic planning. Transportation Science, 39, no. 4 (November): 446–450. Translation of the original 1968 paper “Uber ein Paradoxon aus der Verkehrsplanung” into English by Anna Nagurney and Tina Wakolbinger.

Brian Hayes. 2015a. Traffic Jams in Javascript. bit-player blog entry, posted 18 June 2015.

Brian Hayes. 2015b. Playing in TrafficAmerican Scientist, 103 (July–August): 260–263.

Gina Kolata. 1990. What if they closed 42d Street and nobody noticed? New York Times, December 25, 1990, page 38.

Anna Nagurney and David Boyce. 2005. Preface to “On a Paradox of Traffic Planning”. Transportation Science, 39, no. 4 (November): 443–445.

Anna Nagurney and Ladimer Nagurney. ForthThe Braess Paradox. In International Encyclopedia of Transportation, edited by B. Noland, R. Vickerman, and Dick Ettema. Elsevier. Invited chapter, pre-print available at https://supernet.isenberg.umass.edu/articles/braess-encyc.pdf.

John G. Wardrop. 1952. Some theoretical aspects of road traffic research. Proc. Institution Civil Engineers, Part II, 1, no. 2 (August): 325–378.

# Topology from Analysis: A Mini-Primary Source Project for Topology Students

Author(s):
Nicholas A. Scoville (Ursinus College)

During my second year at Ursinus, I taught an undergraduate course in point-set topology. My doctoral work was in topology, so I was quite excited to teach the course. We were almost halfway though the semester, and I thought the class was going great. “These kids are really getting it," I thought to myself. Yet one day in class while discussing a proof involving the product topology, a student remarked that “I can follow the proof and I understand it, but I don’t know what we are doing." What the student later told me this meant was that he could justify every line of the proof, but didn’t understand its significance. Great, now we know that the collection of all preimages of open sets under the projection forms a subbasis . . . so? This student’s experience was not unlike my own as an undergraduate. When I took point-set topology, I knew that what we were doing was math, but I could not tell you how concepts in topology would naturally arise from the other math that I had studied. The mini-Primary Source Project Topology from Analysis is an attempt to bridge this conceptual gap between topology and more familiar concepts in mathematics.

 The project is based on the 1872 paper “Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen” (On the Extension of a Theorem from the Theory of Trigonometric Series) by Georg Cantor (1845–1918). A year earlier, Cantor had published his proof that a Fourier series representation for a function is unique even when the function is not defined on a finite set (Cantor 1871). His 1872 paper extended that result by showing that uniqueness holds even when the function is not defined on certain kinds of infinite sets, sets which he called “point sets of type n." These point sets had the property that their (n + 1)st derived set, obtained by iterating the process of collecting together the limit points of a set to form a new set, is empty. While such sets are sets of real numbers, the main property they satisfy has less to do with the distances between the real numbers they contain from one another, and more to do with a global property of the set as a whole. Namely, given a set of real numbers, if we take the limit points of the limit points of the limit points of the limit points ad infinitum, do we ever end up with the empty set? This purely topological concept, then, arises naturally from analysis, thereby making it more familiar and grounded to the student. Georg Cantor (1845–1918) Convergence Portrait Gallery

The student project Topology from Analysis begins by investigating what the question that interested Cantor means. By closely examining an example of an infinite set on which a function could be undefined but still have a unique Fourier series representation, the student is able to see why someone would write down the definition of a limit point. That same example also lays out the basic idea behind Cantor’s main result. Cantor’s definition of a derived set and its subsequent iterations are then introduced, to set up the notion that some infinite sets (e.g., point sets of finite type) are “more well-behaved" than others. Following some practice in computing derived sets, the project ends by guiding the student through Cantor’s proof of the main result of his 1872 paper. By tracing Cantor’s proof in this way, students are more naturally led to see the motivation behind the technique of proof by induction, as well as sophisticated topological concepts such as limit point and derived set.

Diagram from Cantor's 1872 proof for uniqueness of Fourier series representation,
illustrating the relationship of points on an interval in key step of his proof.
See also the author's Convergence article, “Georg Cantor at the Dawn of Point-Set Topology.”

The complete project Topology from Analysis (pdf) is ready for student use, and the LaTeX source code is available from the author by request. A set of instructor notes that explain the purpose of the project and guide the instructor through the goals of each of the individual sections is appended at the end of the student project. This project is the sixteenth in A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources appearing in Convergence, for use in courses ranging from first-year calculus to analysis, number theory to topology, and more. Links to other mini-PSPs in the series appear below, including six additional mini-PSPs on topics in undergraduate-level analysis and topology. The full TRIUMPHS collection offers a total of six mini-PSPs and two more extensive “full-length” PSPs for use with students of topology.

Acknowledgments

The development of the student project Topology from Analysis has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project with funding from the National Science Foundation’s Improving Undergraduate STEM Education Program under Grants No. 1523494, 1523561, 1523747, 1523753, 1523898, 1524065, and 1524098. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily reflect the views of the National Science Foundation. The author gratefully acknowledges this support, with special thanks to TRIUMPHS PI Janet Heine Barnett, who provided assistance with the historical content of the project.

Reference

G. Cantor. 1871. ÜberMath. Ann., 4:139–143.

G. Cantor. 1872. Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen. Math. Ann., 5(1):123–132.

N. Scoville. 2012. Georg Cantor at the Dawn of Point-Set Topology. Convergence, 9. DOI:10.4169/loci003861.