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Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS)

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• F1: A Genetic Context for Understanding the Trigonometric Functions
Author: Danny Otero
Description: In this project, we explore the genesis of the trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. The goal is to provide the typical student in a pre-calculus course some context for understanding these concepts that is generally missing from standard textbook developments. Trigonometry emerged in the ancient Greek world (and, it is suspected, independently in China and India as well) from the geometrical analyses needed to solve basic astronomical problems regarding the relative positions and motions of celestial objects. While the Greeks (Hipparchus, Ptolemy) recognized the usefulness of tabulating chords of central angles in a circle as aids to solving problems of spherical geometry, Hindu mathematicians, like Varahamahira (505--587), found it more expedient to tabulate half-chords, whence the use of the sine and cosine became popular. We will examine an excerpt from this work, wherein Varahamahira describes a few of the standard modern relationships between sine and cosine in the course of creating a sine table. In the 11th century, the Arabic scholar and expert on Hindu science Abu l-Rayhan Muhammad al-Biruni (973--1055) published The Exhaustive Treatise on Shadows (ca.~1021). In this work, we see how Biruni presents geometrical methods for the use of sundials; the relations within right triangles made by the gnomon of a sundial and the shadow cast on its face lead to the study and tabulation of values of the tangent and cotangent, secant and cosecant. Biruni also works out the relationships that these quantities have with the sines and cosines of the angles. However, the modern terminology for the standard trigonometric quantities is not established until the European Renaissance. Foremost in this development is the landmark On Triangles (1463) by Regiomontanus (Johannes Muller). Regiomontanus exposes trigonometry in a purely geometrical form and then applies the ideas to problems in circular and spherical geometry. We examine a few of the theorems that explore the trigonometric relations and which are used to solve triangle problems.

• F2: Determining the Determinant
Author: Danny Otero
Description: This project in linear algebra illustrates how the mathematicians of the eighteenth and nineteenth centuries dealt with solving systems of linear equations in many variables, a complicated problem that ultimately required attention to issues of the notation and representation of equations as well as careful development of the auxiliary notion of a "derangement'' or "permutation." Colin Maclaurin (1698-1746) taught a course in algebra at the University of Edinburgh in 1730 whose lecture notes include formulas for solving systems of linear equations in 2 and 3 variables; an examination of these lecture notes illustrate the forms of the modern determinant long before the notion was formally crystalized. In 1750, Gabriel Cramer (1704-1752) published his landmarkIntroduction a l'Analyse des Lignes Courbes algebriques (Introduction to the Analysis of Algebraic Curves). In an appendix to this work, Cramer tackles the solution of linear systems more systematically, providing a formula for the solution to such a system, today known as Cramer's Rule. More significantly, he points out the rules for formation of the determinantal expressions that appear in the formulas for the solution quantities, using the term "derangement" to refer to the complex permuting of variables and their coefficients that gives structure to these expressions. These ideas reach maturity in an 1812 memoir by Augustin-Louis Cauchy (1789-1857) entitled Memoire sur les functions qui ne peuvent obtenir que deux valeurs egales et de signes contraires par suite des transpositions operees entre les variables quelles renferement (Memoir on those functions which take only two values, equal but of opposite sign, as a result of transpositions performed on the variables which they contain). In this work, Cauchy provides a full development of the determinant and its permutational properties in an essentially modern form. Cauchy uses the term "determinant" (adopted from Gauss) to refer to these expressions and even adopts an early form of matrix notation to express the formulas for solving a linear system.

• F3: Solving a System of Linear Equations Using Ancient Chinese Methods
Author: Mary Flagg
Description: Gaussian elimination for solving systems of linear equations is one of the first topics in a standard linear algebra class. The algorithm is named in honor of Carl Friedrich Gauss (1777-1855), but the technique was not his invention. In fact, Chinese mathematicians were solving linear equations with a version of elimination as early as 100 AD. This project has the students study portions of Chapter 8 Rectangular Arrays in The Nine Chapters on the Mathematical Art to learn the technique known to the Chinese by 100 AD. Then students will then read the commentary to Chapter 8 of Nine Chapters given by Chinese mathemtician Liu Hui in 263 AD and be asked how his commentary helps understanding. The method of the Nine Chapters will be compared to the modern algorithm. The similarity between the ancient Chinese and the modern algorithm exemplifies the sophisticated level of ancient Chinese mathematics. The format of the Nine Chapters as a series of practical problems and solutions reinforces the concept that mathematics is connected to everyday life.

• F4: Investigating Difference Equations.
Author: Dave Ruch
Description: Abraham de Moivre is generally given credit for the first systematic method for solving a general linear difference equation with constant coefficients. He did this by creating and using a general theory of recurrent series. While de Moivre's methods are accessible to students in a sophomore/junior discrete math course, they are not as clear or straightforward as the methods found in today's textbooks. Building on de Moivre's work, Daniel Bernoulli published a 1728 paper in which he laid out a simpler approach, along with illuminating examples and a superior exposition. The first part of the project develops de Moivre's approach with excerpts from original sources. The second part will give Bernoulli's 1728 methodology, no doubt more attractive to most students. Ideally this project will help students understand and appreciate how mathematics is developed over time, in addition to learning how to solve a general linear difference equation with constant coefficients. Sources include: De Moivre's 1718 Doctrine of Chances, Daniel Bernoulli's 1728 paper (translated) "Observations about series produced by adding or subtracting their consecutive terms which are particularly useful for determining all the roots of algebraic equations."

• F5: Quantifying certainty: the p-value
Author: Dominic Klyve
Description: The history of statistics is closely linked to our ability to quantify uncertainty in predictions based on partial information. In modern statistics, this rather complex idea is crystallized in one concept: the p-value. Understanding p-values is famously difficult for students, and statistics professors often have trouble getting their students to understand the rather precise nuances involved in the definition. In this project, students will work to build a robust understanding of p-values by working through some early texts on probability and certainty. These will include the Sir Ronald Fisher's famous Statistical Methods for Research Workers, and will also cover earlier attempts that came very close to the modern concept, such as Buffon's Essai d'Arithmetique Morale.

• F6: The Exigency of the Parallel Postulate.
Author: Jerry Lodder
Description: In this project, we examine the use of the parallel postulate for such basic constructions as the distance formula between two points and the angle sum of a triangle (in Euclidean space). Beginning with Book I of Euclid's (ca. 300 b.c.) Elements, we witness the necessity of the parallel postulate for constructing such basic figures as parallelograms, rectangles and squares. This is followed by Euclid's demonstration that parallelograms on the same base and between the same parallels have equal area, an observation essential for the proof of the Pythagorean Theorem. Given a right triangle, Euclid constructs squares on the three sides of the triangle, and shows that the area of the square on the hypotenuse is equal to the combined area of the squares on the other two sides. The proof is a geometric puzzle with the pieces found between parallel lines and on the same base. The project will stress the ancient Greek view of area, which greatly facilitates an understanding of the Pythagorean Theorem. This theorem is then essential for the modern distance formula between two points, often used in high school and college mathematics, engineering and science courses. During the Autumn Semester of 2013, PI Jerry Lodder used a preliminary version of this project in an upper divisional geometry course for secondary education majors and mathematics majors. Several students wrote that they had always thought of the Pythagorean Theorem as an equation in algebra, and for the first time, understood the geometry behind this theorem. Other students wrote that they liked geometry proofs, because they could see what was happening, and would cut squares out of construction paper to prove the Pythagorean Theorem for the high school classes that they would teach. The project is designed for courses in geometry taken both by mathematics majors and secondary education majors.

• F7: The Failure of the Parallel Postulate.
Author: Jerry Lodder
Description: This project will develop the non-Euclidean geometry pioneered by Janos Bolyai (1802-1860), Nikolai Lobachevsky (1792-1856) and Carl Friedrich Gauss (1777 - 1855). Beginning with Adrien Marie Legendre's (1752 - 1833) failed proof of the parallel postulate, the project will begin by questioning the validity of the Euclidean parallel postulate and the consequences of doing so. How would distance be measured without this axiom, how would "rectangles" be constructed, and what would the angle sum of a triangle be? The project will continue with Lobachevsky's work, where he states that in the uncertainty whether there is only one line though a given point parallel to a given line, he considers the possibility of multiple parallels, and continues to study the resulting geometry, limiting parallels, and proprieties of triangles in this new world. This will be followed by a discussion of distance in hyperbolic geometry from the work of Bolyai and Lobachevksy. The project will show that all triangles in hyperbolic geometry have angle sum less than 180 degrees, with zero being the sharp lower bound for such a sum, as anticipated by Gauss. The project will continue with the unit disk model of hyperbolic geometry provided by Henri Poincare (1854 - 1912), and, following the work of Albert Einstein (1879- 1955), close with the open question of whether the universe is best modeled by Euclidean or non-Euclidean geometry.

• F8: Richard Dedekind and the Creation of an Ideal: Early Developments in Ring Theory
Author: Janet Barnett
Description: As with other structures in modern Abstract Algebra, the ring concept has deep historical roots in several nineteenth century mathematical developments, including the work of Richard Dedekind (1831-1916) on algebraic number theory. This project draws on Dedekind's 1877 text Theory of Algebraic Numbers as a means to introduce students to the elementary theory of commutative rings and ideals. Characteristics of Dedekind's work that make it an excellent vehicle for students in a first course on abstract algebra include his emphasis on abstraction, his continual quest for generality and his careful methodology. The 1877 version of his ideal theory (the third of four versions he developed in all) is an especially good choice for students to read, due to the care Dedekind devoted therein to motivating why ideals are of interest to mathematicians by way of examples from number theory that are readily accessible to students at this level. The project begins with Dedekind's discussion of several specific integral domains, including the example of the integers adjoin 5i, which fails to satisfy certain expected number theoretic properties (e.g. a prime divisor of a product should divide one of the factors of that product). Having thus set the stage for his eventual introduction of the concept of an ideal, the project next offers students the opportunity to explore the general algebraic structures of a ring, integral domain and fields. Following this short detour from the historical story - rings themselves were first singled out as a structure separate from ideals only in Emmy Noether's later work - the project returns to Dedekind's own exploration of ideals and their basic properties. Starting only with his formal definition of an ideal, project tasks lead students to explore the basic concept of and elementary theorems about ideals (e.g., the difference between ideals and subrings, how properties of subrings and ideals may differ from the properties of the larger ring, properties of ideals in rings with unity). Subsequent project tasks based on excerpts from Dedekind's study of principal ideals and divisibility relationships between ideals conclude with his (very modern!) proofs that the least common multiple and the greatest common divisor of two ideals are also ideals. The project closes by returning to Dedekind's original motivation for developing a theory of ideals, and considers the sense in which ideals serve to recover the essential properties of divisibility - such as the fact that a prime divides a product of two rational integer factors only if it divides one of the factors - for rings that fail to satisfy these properties. No prior familiarity with ring theory is assumed in the project. Although some familiarity with elementary group theory can be useful in certain portions of the project, it has also been successfully used with students who had not yet studied group theory. For those who have not yet studied group theory (or those who have forgotten it!), basic definitions and results about identities, inverses and subgroups are fully stated when they are first used within the project (with the minor exception of Lagrange's Theorem for Finite Groups which is needed in one project task). The only number theory concepts required should be familiar to students from their K-12 experiences; namely, the definitions (within the integers) of prime, composite, factor, multiple, divisor, least common multiple, and greatest common divisor

• F9: Primes, divisibility, and factoring
Author: Dominic Klyve
Description: Questions about primality, divisibility, and the factorization of integers have been part of mathematics since at least the time of Euclid. Today, they comprise a large part of an introductory class in number theory, and they are equally important in contemporary research. In this project, students investigate the development of the modern theory of these three topics. Looking briefly at the writings of Euclid and Fermat, we shall then turn to the primary focus of the paper -- a remarkable 1732 paper by Leonhard Euler. This, Euler's first paper in number theory, contains a surprising number of new ideas in the theory of numbers. In a few short pages, he provides for the first time a factorization of 2^{2^5}+1 (believed by Fermat to be prime), discusses the factorization of 2^n-1 and 2^n+1, and begins to develop the ideas which would lead to the first proof of what we now call Fermat's Little Theorem. In this work, Euler provides few proofs. By providing these, students develop an intimacy with the techniques of number theory, and simultaneously come to discover the importance of modern ideas and notation in the field.

• F10: The Pell Equation in India
Author: Keith Jones and Toke Knudsen
Description: The Pell equation is the Diophantine equation

x2 - Ny2 = 1

where N is a non-square, positive integer. The equation has infinitely many solutions in positive integers x and y, though finding a solution is not trivial.

In modern mathematics, the method of solving the Pell equation via continued fractions was developed by Lagrange (1736-1813). However, much earlier, Indian mathematician made significant contributions to the study of the Pell equation and its solution. Brahmagupta (b. 598 CE) discovered that the Pell equation can be solved if a solution to

x2 - Ny2 = k

where k=-1,2,-2,4,-4 is known. Later a method, a cyclic algorithm known in Sanskrit as cakravala, to solve the Pell equation was developed by Jayadeva and Bhaskara ii (b. 1114 CE).

• F11: Greatest Common Divisor: Algorithm and Proof
Author: Mary Flagg
Description: Finding the greatest common divisor of two integers is a foundational skill in mathematics, needed for tasks from simplifying fractions to cryptography. Yet, the best place to look for a simple algorithm for finding the greatest common divisor is not in a modern textbook, but in the writings of the ancient Chinese and the Elements of Euclid in ancient Greece. In this project we will discover how the mutual subtraction algorithm evolved in ancient China, from an text dated ca. 200 BCE to the version in The Nine Chapters on the Mathematical Art to the explanation of the algorithm by Liu Hui in 263 CE. We will then discover the algorithm of Euclid ca. 300 BCE and examine his careful proof. Parallel to the story of the development of the algorithm is a beautiful illustration of the history of proof. Proof in ancient China was not based on prepositional logic, but on demonstrating the correctness of an algorithm. Euclid is the pioneer of logical proof, yet his proof has flaws when examined in the light of modern rigor. Therefore, the project finishes by explicitly stating the properties of integers assumed in the proof of Euclid, and justifying the correctness of Euclid's iterative method using the power of inductive proof.

• F12: The Mobius Inversion Formula
Author: Carl Lienert
Description: It is often easier to find a formula for the divisor sum of an arithmetic function, f(n), than it is to directly find a formula for f(n). Mbius Inversion can then be used to find a formula for f(n) itself. The first time you see this in action it's as cool as the first time you see Mobius' more well-known, but equally cool, Mobius strip. A typical first application of his inversion formula in a number theory class is to find a formula for Euler's phi function, the number of integers between 1 and n relatively prime to n. But, how and why did Mobius develop this technique and the associated Mobius function? In this project, we'll read Mobius' Uber eine besondere Art von Umkehrung der Reihen from 1832 to find out. We'll also study the applications Mobius provides. Finally, we'll make the connection to the modern presentation of the technique.

• F13: Bolzano on Continuity and the Intermediate Value Theorem
Author: Dave Ruch
Description: The foundations of calculus were not yet on firm ground in early 1800's. Students will read from Bernard Bolzano's 1817 paper, in which he gives a definition of continuity and formulates his version of the least upper bound property of the real numbers. Students will then read Bolzano's proof of the Intermediate Value Theorem.

• F14: Rigorous Debates over Debatable Rigor: Monster Functions in Introductory Analysis
Author: Janet Barnett
Description: Although the majority of concepts studied have already been encountered in the light of their first-year calculus courses, students in an introductory analysis course are now required to re-examine these concepts through a new set of powerful lenses. Among the new creatures revealed by these lenses are a certan family of functions. In the late nineteenth century, Gaston Darboux (1842--1917) and Giuseppe Peano (1858--1932) each used members of this function family to critique the level of rigor in certain contemporaneous proofs. Reflecting on the introduction of such functions into analysis for this purpose, Henri Poincare (1854--1912) lamented: "Logic sometimes begets monsters. The last half-century saw the emergence of a crowd of bizarre functions, which seem to strive to be as different as possible from those honest [honnetes] functions that serve a purpose. No more continuity, or continuity without differentiability, etc. What's more, from the logical point of view, it is these strange functions which are the most general, [while] those which arise without being looked for appear only as a particular case. They are left with but a small corner. In the old days, when a new function was invented, it was for a practical purpose; nowadays, they are invented for the very purpose of finding fault in our father's reasoning, and nothing more will come out of it." Yet Emile Borel (1871--1956) proposed two reasons why these "refined subtleties with no practical use" should not be ignored: "[O]n the one hand, until now, no one could draw a clear line between straightforward and bizarre functions; when studying the first, you can never be certain you will not come across the others; thus they need to be known, if only to be able to rule them out. On the other hand, one cannot decide, from the outset, to ignore the wealth of works by outstanding mathematicians; these works have to be studied before they can be criticized." In this project, students come to know these "monster" functions directly from the writings of the influential French mathematician Darboux and one of the mathematicians whose works he critiqued, Guillaume Houel (1823--1886). Project tasks based on the sources prompt students to refine their intuitions about continuity, differentiability and their relationship, and also includes an optional section that introduces them to the concept of uniform differentiability. The project closes with an examination of Darboux's proof of the theorem that now bears his name: every derivative has the intermediate value property. The project thus fosters students' ability to read and critique proofs in modern analysis, thereby enhancing their understanding of current standards of proof and rigor in mathematics more generally.

• F15: An Introduction to the Algebra of Complex Numbers and the Geometry in the Complex Plane
Author: Diana White and Nicholas Scoville
Description: In this project, we will study the basic definitions, as well as geometric and algebraic properties, of complex numbers via Wessel's 1797 paper On the Analytical Representation of Direction. An attempt Applied Chiefly to Solving Plane and Spherical Polygons, the first to develop the geometry of complex numbers.

• F16: Nearness without distance
Author: Nicholas Scoville
Description: Point--set topology is often described as nearness without distance." Although this phrase is intended to convey some intuitive notion of the study of topology, the student is often left feeling underwhelmed after seeing this idea made precise in the definition of a topology. This project, will develop topology, starting with a question in analysis, into a theory of nearness of points that took place over several decades. Motivated by a question of uniqueness of a Fourier expansion, Cantor (1845--1918) develops a theory of nearness based on the notion of limit points over several papers beginning in 1872 and lasting over a decade . Borel then takes Cantor's ideas and begins to apply them to a more general setting, Finally, Hausdorff's (1868--1842) develops a coherent theory of topology in his famous 1914 book Grundz\"{u}ge der Mengenlehre. The purpose of this project is to introduce the student to the ways in which we can have nearness of points without a concept of distance by studying the works of Cantor, Borel, and Hausdorff.

• F17: Connectedness-- its evolution and applications
Author: Nicholas Scoville
Description: The need to define the concept of connected" is first seen in an 1883 work of Cantor (1845--1918) where he gives a rigourous definition of a continuum. After its inception by Cantor, definitions of connectedness were given by Jordan (1838--1922) and Schoenflies (1853--1928), among others, culminating with the current definition proposed by Lennes (1874--1951) in 1905 . This led to connectedness being studied for its own sake by Knaster and Kuratowski. In this project, we will trace the development of the concept of connectedness through the works of these authors, proving many fundamental properties of connectedness along the way.

• F18: Construction of the Figurate Numbers.
Author: Jerry Lodder
Description: This project will be accessible to a wide audience, requiring only arithmetic and elementary high school algebra as a prerequisite. The project will open by studying the triangular numbers, which enumerate the number of dots in regularly shaped triangles, forming the sequence 1, 3, 6, 10, 15, 21, etc. Student activities will include sketching certain of these triangles, counting the dots, and studying how the $n$th triangular number, $T_n$, is constructed from the previous triangular number, $T_{n-1}$. Further exercises will focus on tabulating the values of $T_n$, conjecturing an additive pattern based on the first differences $T_n - T_{n-1}$, and conjecturing a multiplicative pattern based on the quotients $T_n/n$. The triangular numbers will be related to probability by enumerating the number of ways two objects can be chosen from $n$ (given that order does not matter). Other sequences of two-dimensional numbers based on squares, regular pentagons, etc. will be studied from the work of Nicomachus. The project will continue with the development of the pyramidal numbers, $P_n$, which enumerate the number of dots in regularly shaped pyramids, forming the sequence 1, 4, 10, 20, 35, etc. Student activities will again include sketching certain of these pyramids, tabulating the values of $P_n$, conjecturing an additive pattern based on the first differences $P_n - P_{n-1}$, and conjecturing a multiplicative pattern based on the quotients $P_n/T_n$. The pyramidal numbers will be related to probability by counting the number of ways three objects can be chosen from $n$. Similar exercises will be developed for the four-dimensional (triangulo-triangular) numbers and the five-dimensional (triangulo-pyramidal) numbers. The multiplicative patterns for these figurate numbers will be compared to those stated by Pierre de Fermat (1601--1665), such as The last number multiplied by the triangle of the next larger is three times the collateral pyramid'' , which, when generalized, hint at a method for computing the $n$-dimensional figurate numbers similar to an integration formula.

• F19: Pascal's Triangle and Mathematical Induction.
Author: Jerry Lodder
Description: In this project students will build on their knowledge of the figurate numbers gleaned in the previous project. The material will be centered around excerpts from Blasie Pascal's (1623--1662) Treatise on the Arithmetical Triangle'' \cite{Pascal}, in which Pascal employs a simple organizational tool by arranging the figurate numbers into the columns of one table. The $n$th column contains the $n$-dimensional figurate numbers, beginning the process with $n = 0$. Pascal identifies a simple principle for the construction of the table, based on the additive patterns for the figurate numbers. He then notices many other patters in the table, which he calls consequences of this construction principle. To verify that the patterns continue no matter how far the table is constructed, Pascal states verbally what has become known as mathematical induction. Students will read Pascal's actual formulation of this method, discuss its validity, and compare it to other types of reasoning used in the sciences and humanities today. Finally, students will be asked to verify Pascal's twelfth consequence, where he identifies a pattern in the quotient of two figurate numbers in the same base of the triangle. This then leads to the modern formula for the combination numbers (binomial coefficients) in terms of factorials.

• F21: An Introduction to a Rigorous Definition of Derivative.
Author: Dave Ruch
Description: This project is designed to introduce the derivative with some historical background from Newton, Berkeley and L'Hopital. Cauchy is generally credited with being among the first to define and use the derivative in a near modern fashion. Students will read his definition with examples from and explore relevant examples and basic properties.

• F22: Investigations into Bolzano’s Bounded Set Theorem
Author: Dave Ruch
Description: Bernard Bolzano was among the first mathematicians to rigorously analyze the completeness property of the real numbers .This project investigates his formulation of the least upper bound property from his 1817 paper. Students will read his proof of a theorem on this property. Bolzano's proof also inspired Karl Weierstrass decades later in his proof of what is now known as the Bolzano-Weierstrass Theorem.

• F23: The Mean Value Theorem
Author: Dave Ruch
Description: The Mean Value Theorem has come to be recognized as a fundamental result in a modern theory of the differential calculus. Students will read from Cauchy's e orts in to rigorously prove this theorem for a function with continuous derivative. Later in the project students explore a very different approach some forty years later by mathematicians Serret and Bonnet.

• F24: Abel and Cauchy on a Rigorous Approach to Infinite Series
Author: Dave Ruch
Description: Infinite series were of fundamental importance in the development of the calculus. Questions of rigor and convergence were of secondary importance early on, but things began to change in the early 1800's. When Niels Abel moved to Paris in 1826, he was aware of many paradoxes with infinite series and wanted big changes. In this project, students will read from Cauchy's 1821 Cours d'Analyse, in which he carefully defines infinite series and proves some properties. Students will then read from Abel's paper in which he attempts to correct a flawed series convergence theorem from Cauchy's book.

• F25: The Definite Integrals of Cauchy and Riemann
Author: Dave Ruch
Description: Rigorous attempts to define the definite integral began in earnest in the early 1800's. One of the pioneers in this development was A. L. Cauchy (1789-1857). In this project, students will read from his 1823 study of the definite integral for continuous functions. Then students will read from Bernard Riemann's 1854 paper, in which he developed a more general concept of the definite integral that could be applied to functions with infinite discontinuities.

• F26: Gaussian Integers and Dedekind's Creation of an Ideal: A Number Theory Project
Author: Janet Barnett
Description: In the historical development of mathematics, the nineteenth century was a time of extraordinary change during which the discipline became more abstract, more formal and more rigorous than ever before. Within the subdiscipline of algebra, these tendencies led to a new focus on studying the underlying structure of various number (and number-like) systems related to the solution of various equations. The concept of a group, for example, was singled out by Evariste Galois (1811-1832) as an important algebraic structure related to the problem of finding all complex solutions of a general polynomial equation. Two other important algebraic structures - ideals and rings - emerged later in that century from the problem of finding all integer solutions of various equations in number theory. In their efforts to solve these equations, nineteenth century number theorists were led to introduce generalizations of the seemingly simple and quite ancient concept of an integer. This project examines the ideas from algebraic number theory that eventually led to the new algebraic concepts of an ideal' and a ring' via excerpts from the work of German mathematician Richard Dedekind (1831-1916). A key feature of Dedekind's approach was the formulation of a new conceptual framework for studying problems that were previously treated algorithmically. Dedekind himself described his interest in solving problems through the introduction of new concepts as follows: "The greatest and most fruitful progress in mathematics and other sciences is through the creation and introduction of new concepts; those to which we are impelled by the frequent recurrence of compound phenomena which are only understood with great difficulty in the older view." In this project, students encounter Dedekind's creative talents first hand through excerpts from his 1877 Theory of Algebraic Integers. The project begins with Dedekind's description of the number theoretic properties of two specific integral domains: the set of rational integers and the set of Guassian integers. The basic properties of Gaussian integer divisibility are then introduced, and connections between Gaussian Primes and number theory results such as The Two Squares Theorem are explored. The project next delves deeper into the essential properties of rational primes in the integers --- namely, the Prime Divisibility Property and Unique Factorization - to see how these are mirrored by properties of the Guassian Primes in the Gaussian Integers. Concluding sections of the project then draw on Dedekind's treatment of indecomposables in the integral domain integers adjoin the square root of 5, in which Prime Divisibility Property and Unique Factorization both break down, and briefly consider the mathematical after-effects of this break down' in Dedekind's creation of an ideal.

• F27: Otto Holder's Formal Christening of the Quotient Group Concept
Author: Janet Barnett
Description: Today's undergraduate students are typically introduced to quotient groups only after meeting the concepts of equivalence, normal subgroups and cosets. Not surprisingly, the historical record reveals a different course of development. Although quotient groups implicitly appeared in Galois' work on algebraic solvability in the 1830's, that work itself pre-dated the development of an abstract group concept. Even Cayley's 1854 paper in which a definition of an abstract group first appeared was premature, and went essentially ignored by mathematicians for decades. Permutation groups were extensively studied during that time, however, with implicit uses of quotient groups naturally arising within it. Camille Jordan, for example, used the idea of congruence of group elements modulo a subgroup to produce a quotient group structure . Thus, when Otto Holder gave what is now considered to be the first "modern" definition of quotient groups in 1889, he was able to treat the concept as neither new nor difficult. This project will follow the evolution of the concept of abstract quotient groups within the context of the work done by Jordan and others who paved the way for Holder, and then examine Holder's own treatment of the quotient group concept.

• F28: The Roots of Early Group Theory in the Works of Lagrange
Author: Janet Barnett
Description: This project studies works by one of the early precursors of abstract group, French mathematician J. L. Lagrange (1736-1813). An important figure in the development of group theory, Lagrange made the first real advance in the problem of solving polynomial equations by radicals since the work of Cardano and his 16th century contemporaries. In particular, Lagrange was the first to suggest a relation between permutations and the solution of equations of radicals that was later exploited by the mathematicians Abel and Galois. Lagranges description of his search for a general method of algebraically solving all polynomial equations is a model of mathematical research that make him a master well worth reading even today. In addition to the concept of a permutation, the project employs excerpts from Lagrange's work on roots of unity to develop concepts related to finite cyclic groups. Through their guided reading of excerpts from Lagrange, abstract algebra students will encounter his original motivations while develop their own understanding of these important group-theoretic concepts.

• F29: The Radius of Curvature According to Christiaan Huygens
Author: Jerry Lodder
Description: Curvature is a topic in calculus and physics used today to describe motion (velocity and acceleration) of vector-valued functions. Many modern textbooks introduce curvature via a rather opaque definition, namely the magnitude of the rate of change of the unit tangent vector with respect to arc length. Such a definition offers little insight into what curvature was designed to capture, not to mention its rich historical origins. This project offers Christiaan Huygens's (1629--1695) highly original work on the radius of curvature and its use in the construction of an isochronous pendulum clock. A perfect time-keeper, if one could be constructed to operate at sea, would solve the longitude problem for naval navigation during the Age of Exploration. Amazingly Huygens identifies the path of the isochrone as a cycloid, a curve that had been studied intensely and independently during the seventeenth century. To force a pendulum bob to swing along a cycloidal path, Huygens constrains the thread of the pendulum with metal or wooden plates. He dubs the curve for the plates an evolute of the cycloid and describes the evolutes of curves more general than cycloids. Given a curve and a point B on this curve, consider the circle that best matches the curve at B. Suppose that this circle has center A. Segment AB became known as the radius of curvature of the original curve at B, and the collection of all centers A as B varies over the curve form the evolute. Note that the radius of curvature AB is perpendicular to the original curve at B. For an object moving along this curve, AB helps in the identification of the perpendicular component of the force necessary to cause the object to traverse the curve. This is the key insight into the meaning of curvature.

• F31: Cross Cultural Comparisons: The Art of Computing the Greatest Common Divisor
Author: Mary Flagg
Description: Finding the greatest common divisor between two or more numbers is fundamental to basic number theory. There are three algorithms taught to pre-service elementary teachers: finding the largest element in the intersection of the sets of factors of each number, using prime factorization and the Euclidean algorithm. This project has students investigate a fourth method found in The Nine Chapters on the Mathematical Art, an important text in the history of Chinese mathematics that dates from before 100 ce. This project asks students to read the translated original text instructions for finding the gcd of two numbers using repeated subtraction. Then students are asked to compare this method with the other modern methods taught. Students are led to discover that the Chinese method is equivalent to the Euclidean algorithm. The project is well-suited to a basic algebra course for pre-service elementary and middle school teachers.

• F32: A Look at Desargues' Theorem from Dual Perspectives
Author: Carl Lienert
Description: Girard Desargues (1591 - 1661) is often cited as one of the founders of Projective Geometry. Desargues was, at least in part, motivated by perspective drawing and other practical applications. However, this project focuses on Desargues' Theorem from a mathematical point of view. The theorem, which today goes by his name, is central to Modern Projective Geometry. We'll, in fact, start with a modern statement of Desargues' Theorem in order to more quickly appreciate the elegant beauty of the statement. Desargues' own proof of the theorem is, perhaps ironically, buried at the end of a treatise written by a student of his, Abraham Bosse. The primary focus of this project is to understand Desargues' proof of the theorem, that is, from a classical perspective. To achieve this goal we'll read his proof in Bosse, which will require a visit to other results of Desargues in his more famous work on conics, to classical results of Euclid (ca. 300 B.C.E.), and to a result of Menelaus (ca. 100 C.E.) which we find both in Desargues own colorful writings and in those of Ptolemy (ca. 100 C.E.). The project concludes with a view of Desargues' Theorem from a modern perspective. We'll use the work of Jean Victor Poncelet (1788 - 1867) to reexamine Desargues' Theorem with the assumption that parallel lines meet at a point at infinity and with the principle of duality. The development of the project is intended to both convey the geometrical content and help students learn to do math. It is meant to be accessible to students at the "Introduction to Proofs" level. Many of the exercises explicitly go through a read-understand-experiment-prove cycle. Some experience proving theorems in the spirit of Euclid would be helpful, but not absolutely necessary. A few optional exercises (whose answers could easily be found in a modern text) are left more open.

• F33: Completing the Square: From the Beginnings of Algebra
Author: Danny Otero
Description: This project provides a deep understanding of the standard algebraic method of completing the square, the universal procedure for solving quadratic equations, through the reading of selections from The Compendious Book on Calculation by Restoration and Reduction, written in the ninth century in Baghdad by Muhammad ibn Musa al-Khwarizmi (ca.~780-850 CE), better known today as al-Khwarizmi's. At the same time, students will become acquainted with a sense of how algebraic problem solving was successfully carried out in its earliest days even in the absence of symbolic notation, thereby conveying the importance of modern symbolic practices. Future high school mathematics teachers who will be responsible for teaching algebra courses in their own classrooms will be well served by working through this classroom module. It is also suitable for use in a general history of mathematics course as an introduction to the role of early Islamic era mathematics in the development of algebra as a major branch of mathematics, and is of value to instructors of higher algebra courses who are interested in conveying a sense of the early history of the theory of equations to their students.

• F34: Argand's Development of the Complex Plane
Author: Diana White and Nick Scoville
Description: Complex numbers are a puzzling concept for today's student of mathematics. This is not entirely surprising, as complex numbers were not immediately embraced by mathematicians either. Complex numbers showed up somewhat sporadically in works such as those of Cardano (1501--1576), Tartaglia (1499--1557), Bombelli (1526--1572), and Wallis (1616--1703), but a systematic treatment of complex numbers was given in an essay titled \textit{Imaginary Quantities: Their Geometrical Interpretation} , written by Swiss mathematician Jean-Robert Argand (1768--1822). This project studies the basic definitions, as well as geometric and algebraic properties, of complex numbers via Argand's essay.

• F35: Riemann's Development of the Cauchy-Riemann Equations
Author: Dave Ruch
Description: This project examines the Cauchy-Riemann equations (CRE) and some consequences\ from Riemann's perspective, using excerpts from his 1851 Inauguraldissertation. Students work through Riemann's argument that satisfying the CRE is equivalent to the differentiability of a complex function $w=u\left( x,y\right) +iv\left( x,y\right)$ of a complex variable $% z=x+iy$. Riemann also introduces Laplace's equation for the $u$ and $v$ components of $w$, from which students explore some basic ideas on harmonic functions. Riemann's approach with differentials creates some challenges for modern readers, but works nicely at an intuitive level and motivates the standard modern proof that the CRE follow from differentiability. In the final section of the project, students are introduced to the modern definition of derivative and revisit the CRE in this context.

• F36: Gauss and Cauchy on Complex Integration
Author: Dave Ruch
Description: This project begins with an short excerpt from Gauss on the meaning of definite complex integrals and a claim about their path independence. Students then work through Cauchy's detailed development of a definite complex integral, culminating in his parameterized version allowing for evaluation of these integrals. Students then apply Cauchy's parametric form to illustrate Gauss's ideas on path independence for certain complex integrals.

• F37: Representing and Interpreting Data from Playfair
Author: Diana White, River Bond, Joshua Eastes, and Negar Janani
Description: With the proliferation of data in all aspects of our lives, understanding how to present and interpret visual representations is an essential skill for students to develop. Using the seminal work of William Playfair in his Statistical Breviary, this project introduces students to the bar graph, pie chart, and time series graphs, asking them to interpret real data from the late 1700s and early 1800s. Compound bar graphs, compound time series, and visual depictions incorporating both bar graphs and time series graphs are also included. This project is intended for use in an introductory statistics or data science course at the undergraduate level. However, it could also be used in courses for pre-service teachers, mathematics for liberal arts courses, professional development courses/workshops for teachers, or in history of mathematics courses. It is also potentially suitable for use at the high-school level.

• F39: Stitching Dedekind Cuts to Construct the Real Numbers
Author: Michael P. Saclolo
Description: As a fledgling professor and mathematician, Richard Dedekind (1831-1916) was unsatisfied with the lack of foundational rigor with which differential calculus was taught, and in particular, with the way the set of real numbers and its properties were developed and used to prove the most fundamental theorems of calculus. His efforts to rectify this situation resulted in his 1872 monograph Continuity and Irrational Numbers, which was later published (in 1901) in a longer compilation entitled Essays on the Theory of Numbers. This project guides the students through the development of the real numbers through the examination of Dedekind's own words in translation. The real numbers are formed through Dedekind cuts, which are pairs of subsets of the set of rational numbers that represent a real number. The properties of the real numbers emerge out of corresponding properties of the rationals. The project tasks ask the students to interpret, scrutinize and reflect on the source text. They also challenge them to fill in details that Dedekind had decided to leave out.

• M1: Babylonian numeration
Author: Dominic Klyve
Description: Rather than being taught a different system of numeration, students in this project discover one for themselves. Students are given an accuracy recreation of a cuneiform tablet from Nippur with no initial introduction to Babylonian numerals. Unknown to the students, the table contains some simple mathematics -- a list of the first 13 integers and their squares. Their challenge is threefold: 1) to deduce how the numerals represent values, to work out the mathematics on the tablet, and to decide how to write "72''. A small optional extension of the project asks students to compare the good and bad traits of several numeration systems.

• M2: L'Hopital's rule
Author: Danny Otero
Description: Students of the calculus learn quickly that this grand collection of theoretical ideas and problem solving tools that center on the concepts of derivative and integral ultimately find their justification in the careful computation of limits. And while many of the limits students encounter are trivially determined as applications of the continuity of the underlying functions involved, quite a few are not. Indeterminate forms'' are identified as the chief obstacle to the evaluation of such limits, and L'Hopital's Rule is the standard remedy for resolving these forms. This project introduces students to this important Rule, as it appeared in the first book to expose the entirety of the new'' calculus, Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes(Analysis of the Infinitely Small for the Understanding of Curved Lines), published in 1696 by the French nobleman Guillaume Francois Antoine, Marquis de l'Hopital, based on notes he took from private lessons given him by Jakob Bernoulli. Students also see a justification of the Rule, a few of its major variants, and some applications.

• M3: The Derivatives of the Sine and Cosine Functions
Author: Dominic Klyve
Description: Working through the standard presentation of computing the derivative of sin(x) is a difficult task for a first-year mathematics student. Often, explaining "why'' cosine is the derivative of sine is done via ad-hoc handwaving and pictures. Using an older definition of the derivative, Leonhard Euler gives a very interesting and accessible presentation of finding the derivative of sin(x) in his Institutiones Calculi Differentialis. The entire process can be mastered quite easily in a day's class, and leads to a deeper understanding of the nature of the derivative and of the sine function.

• M4: Beyond Riemann Sums
Author: Dominic Klyve
Description: The purpose of this project is to introduce the method of integration developed by Fermat (1601--1665), in which he essentially used Riemann sums, but allowed the width of the rectangles to vary. Students work through Fermat's text, with the goal of better understanding the method of approximating areas with rectangles.

• M5: Fermat's Method for Finding Maxima and Minima
Author: Ken Monks
Description: In his 1636 article "Method for the Study of Maxima and Minima", Pierre de Fermat (1601--1665) proposed his method of adequality for optimization. In this work, he provided a rather cryptic sounding paragraph of instructions regarding how to find maxima and minima. Afterwards, he claimed that "It is impossible to give a more general method." Here, we trace through his instructions and see how it ends up being mostly equivalent to the standard modern textbook approach of taking a derivative and setting it equal to zero.

• M6: Euler's Calculation of the Sum of the Reciprocals of the Squares
Author: Kenneth M. Monks
Description: This project introduces students to $p$-series via Nicole Oresme's proof of the divergence of the harmonic series in his Quaestiones super Geometriam Euclidis, written in approximately 1350. We continue with Euler's 1740 proof in "De summis serierum reciprocarum" that $\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}$, via his infinite product formula for $\sin(s)/s$.

• M7: Braess' Paradox in City Planning: An Application of Multivariable Optimization
Author: Ken Monks
Description: On December 5, 1990, The New York Times published an article titled What if They Closed 42nd Street and Nobody Noticed? Two of the early paragraphs in this article summarize what happened: On Earth Day this year, New York City's Transportation Commissioner decided to close 42nd Street, which as every New Yorker knows is always congested. 'Many predicted it would be doomsday,' said the Commissioner, Lucius J. Riccio. 'You didn't need to be a rocket scientist or have a sophisticated computer queuing model to see that this could have been a major problem." But to everyone's surprise, Earth Day generated no historic traffic jam. Traffic flow actually improved when 42d Street was closed." This very counterintuitive phenomenon, in which the removal of an edge in a congested network actually results in improved flow, is known as Braess' Paradox. This paradox had actually been studied decades earlier not by rocket scientists, but by mathematicians. In the 1968 paper "On a paradox of traffic planning" , Dietrich Braess (1938-- ) described a framework for detecting this paradox in a network. In this project, we see how the examples he provided can be analyzed using standard optimization techniques from a multivariable calculus course.

• M8: The Origin of the Prime Number Theorem
Author: Dominic Klyve
Description: Near the end of the eighteenth century, Adrien-Marie Legendre (1752--1833) and Carl Friederich 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. Yet both Gauss and Legendre managed to provide exactly that: a nice function for estimating the density of primes. 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 this project, students explore how they may have arrived at their conjectures, compare their similar (though not identical) estimates for the number of primes up to x, and examine some of the ideas related to different formulations of the Prime Number Theorem. Using a letter written by Gauss, they then examine the error in their respective estimates.

• M9: How to Calculate pi: Machin's Inverse Tangents
Author: Dominic Klyve
Description: In this mini-PSP, students rediscover the work of John Machin and Leonhard Euler, who used a tangent identity to calculate pi by hand to almost 100 digits.

• M10: How to calculate pi - Buffon
Author: Dominic Klyve
Description: This project explores Buffon's clever experimental method to calculate pi by throwing a needle on a floor on which several parallel lines have been drawn. It is available in two versions, as described below. Basic notions of geometric probability are introduced in both versions of the project.

M10.1: How to calculate pi - Buffon's Needle (Non-Calculus Version)
This version requires some basic trigonometry, but uses no calculus. It is suitable for use with students who have completed a course in pre-calculus or trigonometry.

• M10.2: How to calculate pi - Buffon's Needle (Calculus Version)
This calculus-based version requires the ability to perform integration by parts. It is suitable for use in Calculus 2, capstone courses for secondary teachers and history of mathematics.

Author: Kenneth M. Monks
Description: The idea of approximating a transcendental function by an algebraic one is most commonly taught to today's calculus students via the machinery of power series. However, that idea goes back much much further! In this project, we visit 7th century India, where Bhaskara I (c. 600-c. 680) gave an incredibly accurate approximation to sine using a rational function in his work Mahabhaskariya (Great Book of Bhaskara). Though there is no surviving account of how exactly he came up with the formula, we guide the student through one plausible approach. The more familiar power series formula for the sine function has been attributed to Madhava of Sangamagrama (c. 1350--c. 1425). Though there are no surviving writings from Madhava's own hand, the Kerala school astronomer Kecallur Nilakantha Somayaji (1444--1544) published Madhava's sine series in the Tantrasamgraha in 1501. The student will translate Madhava's formula, as stated in words by Nilakantha Somayaji, into more modern notation to construct the power series for sine. The project concludes by asking the student to apply Taylor's Error Theorem to compare the accuracy of various formulas for sine. First, the student compares the error in Bhaskara's and Madhava's formulas. Second, the student is asked to construct a sine power series centered at $\pi/2$ for comparison with Bhaskara's approximation.

• M12: Fourier's Proof of the Irrationality of e
Author: Kenneth M. Monks
Description: Few topics are as central to the ideas of the calculus sequence as the infinite geometric series formula, the power series for e^x, and arguing via comparison (direct or limit). Joseph Fourier's (1768-1830) short and beautiful proof that e is irrational combines exactly those three ideas! This project walks the student through the first written account of this argument, which appeared in Melanges d'analyse algebrique et de geometrie} by Janot de Stainville (1783 - 1828). The key idea in Fourier's proof was later leveraged by Joseph Liouville (1809-1882) in Sur i'Irrationalite du nombre e=2.718... to show that e^2 is irrational as well. The project uses excerpts from Liouville's work to point students towards the contrasting behavior of square root of 2 (which becomes rational upon squaring) versus e (which does not) as a stepping stone towards the idea a transcendental number.

• M13-15: Gaussian Guesswork
Author: Janet Barnett
Description: Just prior to his 19th birthday, the mathematical genius Fredrich Gauss began a mathematical diary'' in which he recorded his mathematical discoveries for nearly 20 years. Among these discoveries is the the existence of a beautiful relationship between three particular numbers: the ratio of the circumference of a circle to its diameter , a specific value of an elliptic integral; and the Arithmetic-Geometric Mean of 1 and the square root of 2. Like many of his discoveries, Gauss uncovered this particular relationship through a combination of the use of analogy and the examination of computational data, a practice referred to as Gaussian Guesswork'' by historian Adrian Rice in his Math Horizons article Gaussian Guesswork, or why 1.19814023473559220744 ... is such a beautiful number'' . This set of four mini-projects, based on excerpts from Gauss's mathematical diary and other related manuscripts, will introduce students to the power of numerical experimentation via the story of his discovery of this beautiful relationship, while also serving to consolidate student proficiency of the following topics from the traditional Calculus II course:

M13: Gaussian Guesswork: Elliptic Integrals and Integration by Substitution
M14: Gaussian Guesswork: Polar Coordinates, Arc Length and the Lemniscate Curve
M15: Infinite Sequences and the Arithmetic-Geometric Mean

Each of the four mini-PSPs can be used either alone or in conjunction with any of the other three.

• M16: The logarithm of -1.
Author: Dominic Klyve
Description: Understand the behavior of multiple-valued functions can be a difficult mental hurtle to overcome in the early study of complex analysis. Many eighteenth-century mathematicians also found this difficult. This one-day project will look at a selection from the letters of Euler and Jean Le Rond D'Alembert, in which they argued about the value of log(-1). Their argument not only set the stage for the rise of complex analysis, but helped to end a longstanding friendship.

• M17: Why be so Critical? Nineteenth Century Mathematics and the Origins of Analysis
Author: Janet Barnett
Description: The 17th century witnessed the development of calculus as the study of curves in the hands of Newton and Leibniz, with Euler transforming the subject into the study of analytic functions in the 18th century. Soon thereafter, mathematicians began to express concerns about the relation of calculus (analysis) to geometry, as well as the status of calculus (analysis) more generally. The language, techniques and theorems that developed as the result of the critical perspective adopted in response to these concerns are precisely those which students encounter in an introductory analysis course - but without the context that motivated 19th century mathematicians. This project will employ excerpts from Abel, Bolzano, Cauchy and Dedekind as a means to introduce students to that larger context in order to motivate and support development of the more rigorous and critical view required of students for success in an analysis course.

• M18: Topology from Analysis: Making the Connection.
Author: Nick Scoville
Description: Topology is often described as having no notion of distance, but a notion of nearness. How can such a thing be possible? Isn't this just a distinction without a difference? In this project, we will discover the notion of nearness without distance by studying the work of Georg Cantor and a problem he was investigating involving Fourier series. We will see that it is the relationship of points to each other, and not their distances per se, that is a proper view. We will see the roots of topology organically springing from analysis.

• M19: Connecting Connectedness.
Author: Nick Scoville
Description: Connectedness has become a fundamental concept in modern topology. The concept seems clear enough - a space is connected if it is a "single piece." Yet the definition of connectedness we use today was not what was originally written down. As we will see, connectedness is a classic example of a definition that took decades to arrive at. The first such definition of was given by Georg Cantor in an 1872 paper. After investigating his definition, we trace the evolution of the definition of connectedness through the work of Jordan, Schoenflies, and culminating with the modern definition given by Lennes.

• M20: The Cantor Set before Cantor
Author: Nick Scoville
Description: A special construction used in both analysis and topology today is known as the Cantor set. Cantor used this set in a paper in the 1880s. Yet it appeared as early as 1875 in a paper by the Irish mathematician Henry John Stephen Smith (1826 - 1883). Smith, who is best known for the Smith normal form of a matrix, was a professor at Oxford who made great contributions in matrix theory and number theory. In this project, we will explore parts of a paper he wrote titled On the Integration of Discontinuous Functions.

• M21: A compact introduction to a generalized extreme value theorem
Author: Nick Scoville
Description: In a short paper published just one year prior to his thesis, Maurice Frechet gives a simple generalization one what we might today call the Extreme value theorem. This generalization is a simple matter of coming up with the right" definitions in order to make this work. In this mini PSP, we work through Frechet's entire 1.5 page paper to give an extreme value theorem in more general topological spaces, ones which, to use Frechet's newly coined term, are compact.

• M22: From sets to metric spaces to topological spaces
Author: Nick Scoville
Description: One of the significant contributions that Hausdorff made in his 1914 book Grundzuge der Mengenlehre (Fundamentals of Set Theory) was to clearly lay out for the reader the differences and similarities between sets, metric spaces, and topological spaces. It is easily seen how metric and topological spaces are built upon sets as a foundation, while also clearly seeing what is added" to sets in order to obtain metric and topological spaces. In this project, we follow Hausdorff as he builds topology from the ground up" with sets as his starting point.

• M23: The closure operation as the foundation of topology.
Author: Nick Scoville
Description: The axioms for a topology are well established- closure under unions of open sets, closure under finite intersections of open sets, and the entire space and empty set are open. However, in the early 20th century, multiple systems were being proposed as equivalent options for a topology. Once such system was based on the closure property, and it was the subject of Polish mathematician K. Kuratowski's doctoral thesis. In this mini-project, students will work their a proof that today's axioms for a topology are equivalent to Kuratowski's closure axioms by studying excerpts from both Kuratowski and Hausdorff.

• M24: Euler's Rediscovery of e.
Author: Dave Ruch
Description: The famous constant e appears periodically in the history of mathematics. In this project students will read Euler on e and logarithms from his 1748 book Introductio in Analysin Infinitorum. They will use Euler's ideas to justify the modern definition.

• M25: Henri Lebesgue and the Development of the Integral Concept
Author: Janet Barnett
Description: The primary goal of this project is to consolidate students' understanding of the Riemann integral, and its relative strengths and weaknesses. This is accomplished by contrasting the Riemann integral with the Lebesgue integral, as described by Lebesgue himself in a relatively non-technical 1926 paper. A second mathematical goal of this project is to introduce the important concept of the Lebesgue integral, which is rarely discussed in an undergraduate course on analysis. Additionally, by offering an overview of the evolution of the integral concept, students are exposed to the ways in which mathematicians hone various tools of their trade (e.g., definitions, theorems). \smallskip In light of the project's goals, it is assumed that students have studied the rigorous definition of the Riemann integral as it is presented in an undergraduate textbook on analysis. Familiarity with the Dirichlet function is also useful for two project tasks. These tasks also refer to pointwise convergence of function sequences, but no prior familiarity with function sequences is required.

• M26: Generating Pythagorean Triples via Gnomons: available in two versions.
Author: Janet Barnett
Description: This mini-PSP is designed to provide students an opportunity to explore the number-theoretic concept of a Pythagorean triple. Using excerpts from Proclus' Commentary on Euclid's Elements, it focuses on developing an understanding of two now-standard formulas for such triples, commonly referred to as Plato's method' and Pythagoras' method' respectively. The project further explores how those formulas may be developed/proved via figurate number diagrams involving gnomons.

M26.1: Generating Pythagorean Triples via Gnomons: The Methods of Pythagoras and of Plato via Gnomons
In this less open-ended version, students begin by completing tasks based on Proclus' verbal descriptions of the two methods, and are presented with the task of connecting the method in question to gnomons in a figurate number diagram only after assimilating its verbal formulation. This version of the project may be more suitable for use in lower division mathematics courses for non-majors or prospective elementary teachers.

• M26.2: Generating Pythagorean Triples via Gnomons: A Gnomonic Exploration
In this more open-ended version, students begin with the task of using gnomons in a figurate number diagram to first come up with procedures for generating new Pythagorean triples themselves, and are presented with Proclus' verbal description of each method only after completing the associated exploratory tasks. This version of the project may be more suitable for use in upper division courses in number theory and discrete mathematics, or in capstone courses for prospective secondary teachers.

Although more advanced students will naturally find the algebraic simplifications involved in certain tasks to be more straightforward, the only mathematical content pre-requisites are required in either version is some basic arithmetic and (high school level) algebraic skills. The major distinction between the two versions of this project is instead the degree of general mathematical maturity expected. Both versions include an open-ended comparisons and conjectures' ' penultimate section that could be omitted (or expanded upon) depending on the instructor's goals for the course. The student tasks included in other sections of the project are essentially the same in the two versions as well, but differently ordered in a fashion that renders Version M 26.1 somewhat less open-ended than Version M 26.2.

• M27: Seeing and Understanding Data
Author: Beverly Wood and Charlotte Bolch
Description: Modern data-driven decision-making includes the ubiquitous use of visualizations, mainly in the form of graphs or charts. This project explores the parallel development of thinking about data visually and the technological means for sharing data through pictures rather than words, tables, or lists. Students will have the opportunity to consider both the data and the construction methods along with impact that broadening access to data has had on social concerns. Beginning with the 10th century graph hand-drawn in a manuscript, students will experience woodcuts and plates used for printing data displays through to digital typesetting and dynamic online or video-recorded presentations of data. Early uses of bar chart, pie chart, histogram, line chart, boxplot, and stem-and-leaf plot are compared with modern thoughts on graphical excellence.

• M28: Completing the Square: From the Roots of Algebra
Author: Danny Otero
Description: This project seeks to provide a deep understanding of the standard algebraic method of completing the square, the universal procedure for solving quadratic equations, through the reading of selections from The Compendious Book on Calculation by Restoration and Reduction, written in the ninth century in Baghdad by Muhammad ibn Musa al-Khwarizmi (c.~780--850 CE), better known today simply as al-Khwarizmi. Future high school mathematics teachers who will be responsible for teaching algebra courses in their own classrooms will be well-served by working through this classroom module. It is also suitable for use in a general history of mathematics course as an introduction to the role of early Islamic era mathematics in the development of algebra as a major branch of mathematics, and is of value to instructors of higher algebra courses who are interested in conveying a sense of the early history of the theory of equations to their students.

• M29: Euler's Square Root Laws for Negative Numbers
Author: Dave Ruch
Description: Students read excerpts from Euler's Elements of Algebra on square roots of negative numbers and the laws \sqrt{a}\sqrt{b}=\sqrt{ab}, \sqrt{a}/\sqrt{b}=\sqrt{a/b} when a and/or b is negative. While some of Euler's statements initially appear false, students explore how to make sense of the laws with a broader, multivalued interpretation of square roots. This leads naturally to the notion of multivalued functions, an important concept in complex variables.

• M30: Investigations Into d'Alembert's Definition of Limit
Author: Dave Ruch
Description: The modern definition of a limit evolved over many decades. One of the earliest attempts at a precise definition is credited to d'Alembert (1717-1783). This project is designed to investigate the definition of limit for sequences, beginning with d'Alembert's definition and a modern Introductory Calculus text definition.Two versions of this project are available, for very different audiences, as described below.

M30.1: Investigations Into d'Alembert's Definition of Limit - Calculus Version
This version of the project is aimed at Calculus 2 students studying sequences for the first time. In this version, project tasks first lead students through some examples based on d’Alemert’s completely verbal definition. Students are next asked to find examples illustrating the difference between the modern conception of limit and that of d'Alembert. An optional section then examines these differences in a more technical fashion by having students write definitions for each using inequalities and quantifiers.

• M30.2: Investigations Into d'Alembert's Definition of Limit - Real Analysis Version
This longer version of the project is aimed at Real Analysis students. D'Alembert's definition is completely verbal, and project tasks first lead students through some examples and a translation of this definition to one with modern notation and quantifiers. Students are also asked to find examples illustrating the difference between the modern and d'Alembert definitions. This version of the project then investigates two limit properties stated by d'Alembert, including modern proofs of the properties.

• M31: Playfair's Introduction of Bar Graphs and Pie Charts to Represent Data
Authors: Diana White, River Bond, Joshua Eastes, and Negar Janani
Description: With the proliferation of data in all aspects of our lives, understanding how to present and interpret visual representations is an essential skill for students to develop. Using the seminal work of William Playfair in his Statistical Breviary, this project introduces students to the bar graph (including compound bar graphs) and pie chart, asking them to interpret real data from the late 1700s and early 1800s. Students are also exposed to a modern 3-D misleading pie chart. This project is intended to be usable in a single class period. This project is intended for use in an introductory statistics or data science course at the undergraduate level. However, it could also be used in courses for pre-service teachers, mathematics for liberal arts courses, professional development courses/workshops for teachers, or in history of mathematics courses. It is also potentially suitable for use at the high-school level.

• M32: Playfair's Introduction of Time Series to Represent Data
Author: Diana White, River Bond, Joshua Eastes, and Negar Janani
Description: With the proliferation of data in all aspects of our lives, understanding how to present andinterpret visual representations is an essential skill for students to develop. Using the seminal workof William Playfair in his Statistical Breviary, this project introduces students to the timeseries including compound time series, asking them to interpret real data from the late 1700s andearly 1800s. This project is intended to be usable in a single class period. This project is intended for use in an introductory statistics or data science course at the un-dergraduate level. However, it could also be used in courses for pre-service teachers, mathematicsfor liberal arts courses, professional development courses/workshops for teachers, or in history ofmathematics courses. It is also potentially suitable for use at the high-school level.

• M33: Playfair's Novel Visual Displays of Data
Author: Diana White, River Bond, Joshua Eastes, and Negar Janani
Description: With the proliferation of data in all aspects of our lives, understanding how to present andinterpret visual representations is an essential skill for students to develop. Using the seminalwork of William Playfair in his Statistical Breviary, this project exposes students to the visualdisplays of information that combine compound time series and compound bar graphs, asking themto interpret real data from the late 1700s and early 1800s. This project is intended to be usable ina single class period. This project is intended for use in an introductory statistics or data science course at the un-dergraduate level. However, it could also be used in courses for pre-service teachers, mathematicsfor liberal arts courses, professional development courses/workshops for teachers, or in history ofmathematics courses. It is also potentially suitable for use at the high-school level.

• M34: Regression to the mean
Author: Dominic Klyve
Description: Over a century ago, Francis Galton (1822–1911) noted the curious fact that tall parents usuallyhave children shorter than they, and that short parents, in turn, have taller children. This obser-vation was the beginning of what is now called "regression to the mea" – the phenomenon thatextreme observations are generally followed by more average ones. In this project, students engagewith Galton’s original work on the subject, and build an understanding of the underlying causesfor this sometimes non-intuitive phenomenon.

• M35-37: Solving Linear First Order Differential Equations
Description: In its most general form, the Bernoulli equation can be stated as follows: $$\frac{dy}{dx}+P(x)=Q(x)y^r.$$ The problem of solving this linear first order differential equation was first proposed in print in 1695 by Jacob Bernoulli (1655-1705), as a challenge problem in Acta Eruditorum. This series of mini-PSPs examines three solution methods that have become core topics in courses on differential equations, proposed by Johann Bernoulli (1646-1716), Gottfried Leinbiz (1646--1716) and Leonard Euler (1701-1783) respectively:

• M35: Solving First-Order Linear Differential Equations: Gottfried Leibniz' "Intuition and Check" Method
• M36: Solving Linear First Order Differential Equations: Johann Bernoulli’s Variation of Parameters
• M37: Solving Linear First Order Differential Equations: Leonard Euler’s Integrating Factor

• M38: Wronskians and Linear Independence: A Theorem Misunderstood by Many
Description: We can really only explicitly solve higher order (n>1) linear differential equations a_n(x) \frac{d^n y}{dx^n}+ a_{n-1}(x) \frac{d^{n-1} y}{dx^{n-1}}+ \dots+ a_1(x) \frac{dy}{dx} + a_0(x) y = g(x) when the coefficient functions are either constants (the theme of this PSP), or monomials $c_i x^i$ (called Cauchy-Euler equations). In the constant case, the important observation involves the relationship between the above differential equation and the algebraic equation a_n z^n+a_{n-1}z^{n-1} + \dots + a_1 z + a_0 = 0. This relationship was first noted by Euler in an 1739 letter to Johann Bernoulli (though perhaps expectedly, Bernoulli claimed to have already known it). The argument contained in the correspondence is unfortunately incomplete. However in 1743, Euler published a complete classification of the relationship between constant coefficient linear differential equations and polynomials. This PSP works through Euler’s classification, then concludes by revisiting the original correspondence to consider two examples that Euler and Bernoulli attempted to solve.