**INTRODUCTION**

__1__. Letter from Frederick to Voltaire, May 16, 1749: Aldington, *Letters of Voltaire and Frederick the Great*, pp. 195–196.

__2__. Dean, “What Price Algebra.”

__3__. Algebra texts for use in colleges and specialized schools started to appear after 1660 both in France and at the British Royal Navy school of Christ’s Hospital—and elsewhere later. Ellerton, Kanbir, and Clements, “Historical Perspectives on the Purposes of School Algebra”; Ponte and Guimarães, “Notes for a History of the Teaching of Algebra.”

__4__. Paraphrase of Dewey’s words: Dewey, *Art as Experience*, pp. 36–39; Wikipedia, “Art as Experience.”

__5__. Keyser, “The Humanization of the Teaching of Mathematics.”

**CHAPTER 1: NUMERICAL SYMPHONIES**

__1__. Sfard, “On Two Metaphors for Learning and the Dangers of Choosing Just One.”

__2__. Fauvel and Van Maanen, *History in Mathematics Education: The ICMI Study*, p. 36.

__3__. Pinker, “College Makeover: The Matrix, Revisited.”

__4__. Libin, “Symphony.”

__5__. Libin, “Symphony”; Horton, *The Cambridge Companion to the Symphony*, p. 4.

__6__. Stanley, *The Cambridge Companion to Beethoven*, p. 13; Saccenti, Smilde, and Saris, “Beethoven’s Deafness and His Three Styles.”

__7__. Crowther et al., “Mapping Tree Density at a Global Scale.”

__8__. MLB.com, “Batting Average: 2011”; MLB.com, “Batting Average: 1941.”

__9__. Of note, in his early papers on relativity from 1905 to 1907, Einstein used the uppercase letter *V* to stand for the velocity/speed of light. It was only in 1907 that we see an abrupt shift in his notation to *c*. Webb, *Clash of Symbols*, pp. 140–141.

**CHAPTER 2: ART OF MANEUVER**

__1__. Holmes, *The Oxford Companion to Military History*, p. 541.

__2__. United States Marine Corps, *Warfighting*, p. 100.

__3__. “Maneuver,” *Merriam-Webster Dictionary*.

__4__. The great German mathematician Carl Friedrich Gauss is reputed to have used this maneuver to add the numbers from 1 to 100 when he was an elementary school student. For a more detailed discussion of this common tale, see Hayes, “Gauss’s Day of Reckoning.”

__5__. Viète, *The Analytic Art*, p. 32.

__6__. Cardano, *The Rules of Algebra (Ars Magna)*, p. 8.

__7__. Abdul-Jabbar and Knobler, *Giant Steps*, p. 146.

__8__. Weeks, *The Discovery of the Elements*, p. 2. The original passage in German appeared in Winkler’s 1897 article, “Ueber die Entdeckung neuer Elemente im Verlaufe der letzten fünfundzwanzig Jahre und damit zusammenhängendende Fragen.” [On the discovery of new elements in the course of the last twenty-five years and related questions. (Google Translate German to English)] Shakespeare’s quote is from the play *As You Like It* (ca. 1599): “All the world’s a stage, And all the men and women merely players.”

**CHAPTER 3: NUMERICAL FORENSICS**

__1__. In standard graphical (or Cartesian coordinate) language, the ensemble of values (or what we have called the cloud of values) for the simplest types of variable expressions, when sorted a certain way, take the shape of geometrical paths (often called curves). For example, when the input values of *x* are sorted in ascending order (…, –2, –1, 0, 1, 2,…), the path that the variable expression 3*x* + 7 takes on is in the shape of a line in the *x*-*y* Cartesian plane. (Cartesian coordinate systems are not covered in this book.)

__2__. Recorde, *The Whetstone of Witte*, p. 222.

__3__. Heeffer, “Learning Concepts through the History of Mathematics.”

__4__. Saunderson, *Elements of Algebra in Ten Books*, p. 94; Day, *An Introduction to Algebra (Colleges)*, p. 78; Euler, *Elements of Algebra*, p. 188.

__5__. “Bank,” *Merriam-Webster Dictionary*.

__6__. “Table,” *Merriam-Webster Dictionary*.

__7__. The way this rental works is that the cost is $20 per day plus a $30 one-time base fee. Letting *x* be the number of days gives 20*x* for the per-day amount plus the $30 base fee, which yields the expression for the total cost of renting for *x* days as 20*x* + 30 dollars.

__8__. A slightly more appropriate way to deal with this would be to phrase it as the inequality 20*x* + 30 ≤ 900. Though not discussed in this book, solving linear inequalities (which also involve isolating the *x*) is often covered in elementary algebra courses.

**CHAPTER 4: CONVERGING STREAMS AND EMERGING THEMES**

__1__. Kline, *Mathematics and the Physical World*, p. ix.

__2__. Boyer, *History of Analytic Geometry*, p. 59; Mahoney, “The Beginnings of Algebraic Thought in the Seventeenth Century.”

__3__. The thirteenth-century mathematician Jordanus Nemorarius (Jordanus de Nemore), an important mathematician in the history of algebra in Europe, did some early work with the notion of parameters. Unfortunately, it appears that the results of his use of parameters (particularly in the case of quadratic equations) didn’t germinate and was mostly overlooked by his successors, leaving the way open for Viète’s more intensified rediscovery and exploitation to finally take root 300+ years later. Boyer and Merzbach, *A History of Mathematics*, pp. 257–259.

__4__. Katz and Parshall, *Taming the Unknown*, p. 158.

__5__. Unfortunately, Viète was highly and purposefully dismissive of the contributions by medieval Islamic mathematicians, claiming that he was rediscovering algebra as formulated by the Greeks and desecrated by the “barbarians.” The historical record shows that Viète was incorrect and unfair in this assessment and that the work of medieval Islamic mathematicians (such as Al-Khwarizmi, Abu Kamil, Al-Karaji, and others) as well as pre-Renaissance European mathematicians (such as Jordanus, Leonardo de Pisa, and other abbaco masters) was substantial—with some of it critically contributing to the discoveries obtained by sixteenth-century mathematicians, including Viète himself.

**CHAPTER 5: THE RULE OF DARK POSITION**

__1__. *Shay*, *jidhr*, *radix*: Rahman, Street, and Tahiri, *The Unity of Science in the Arabic Tradition*, pp. 93–94. *Cosa*, *coss*: Katz and Pershall, *Taming the Unknown*, pp. 194, 198, 203, 205, 206. The German algebraists use of the word *coss* was an adaptation of the word *cosa* meaning “thing” in Italian. *Yāvattāvat*: Plofker, *Mathematics in India*, pp. 59, 193.

__2__. Recorde, *The Whetstone of Witte*, p. 220.

__3__. Stedall, *A Discourse Concerning Algebra*, pp. 6, 38.

__4__. Mazur and Pesic, “On Mathematics, Imagination & the Beauty of Numbers.”

__5__. The 180 degrees sum rule for the three angles of a triangle holds for triangles in a flat space. If the space on which the triangle sits is itself curved (such as a sphere), then other rules may apply.

__6__. In some cases, both the correct answer and the mistake could be positive, which might make the error harder to detect.

__7__. Al-Khwārizmī, *Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa’l-muqābala*, p. 5.

__8__. Al-Khwārizmī, *Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa’l-muqābala*, p. 5.

__9__. Al-Khwārizmī, *Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa’l-muqābala*, p. 8.

__10__. Whitehead, *An Introduction to Mathematics*, p. 115.

**CHAPTER 6: THE GRAND PLAY**

__1__. Dewey, *John Dewey: The Later Works 1925–1953*, Volume 10, pp. 372–373.

__2__. “Dewey to Talk on ‘Art, Aesthetic Experience,’” *The Harvard Crimson*.

__3__. Dewey, *Art as Experience*, p. 253.

__4__. Dewey, *Art as Experience*, p. 2.

__5__. Dewey, *Art as Experience*, pp. 37, 39–40.

__6__. The Austrian-American physicist Victor Weisskopf gave this description of the joy of insight in the beginning of his 1991 book of the same name. Weisskopf, *The Joy of Insight*, p. vii.

__7__. Adler, *Stella Adler: The Art of Acting*, p. 85.

__8__. “Hold Infinity in the palm of your hand” from the William Blake poem “Auguries of Innocence.”

__9__. Willingham, “What the NY Times Doesn’t Know About Math Instruction.”

__10__. Willingham, *Why Don’t Students Like School?*, pp. 10–13, 63–64.

__11__. Willingham, *Why Don’t Students Like School?*, pp. 63–64.

__12__. Tall and Thomas, “Encouraging Versatile Thinking in Algebra Using the Computer.”

__13__. Doxiadis, “Embedding Mathematics in the Soul,” p. 22.

__14__. I am not advocating for completely avoiding more complicated numbers and maneuvers in problems, but am simply suggesting that if such situations are allowed to dominate the discussion, especially initially, students may more easily lose sight of the algebraic tasks at hand.

__15__. National Education Association of the United States, *Report of the Committee on Secondary School Studies*, p. 108.

__16__. Shapin and Schaffer, *Leviathan and the Air-Pump*.

__17__. Paret, *Clausewitz and the State*, p. 185. Paret cites, as a source for Clausewitz’s statement, a passage from the 1878 biography *Leben des Generals Clausewitz und der Frau Marie von Clausewitz* by Karl Schwartz.

**CHAPTER 7: ALGEBRAIC AWARENESS**

__1__. Grendler, *Renaissance Education Between Religion and Politics*, p. 6; Cohen and Cohen, *Daily Life in Renaissance Italy*, p. 139; Kleinhenz, *Medieval Italy*, pp. 354–355.

__2__. Cohen, “Numeracy in Nineteenth-Century America,” p. 52.

__3__. Harvard was not the leader in this endeavor. Other colonial colleges such as Yale (through President Thomas Clap’s efforts) had decades earlier preceded Harvard in this and other mathematical requirements. Nathaniel Hammond’s algebra text, *The Elements of Algebra in a New and Easy Method*, was used during part of Clap’s tenure and exhibited remarkable exposition for the era, owing to an extended introduction that gave a historical overview of the subject as well as its comparatively gentle dive into the subject. Hammond also provided copious and coordinated examples throughout the book. Cohen, *A Calculating People*, p. 123; Ellerton and Clements, *Rewriting the History of School Mathematics in North America 1607–1861*, p. 78.

__4__. Littlefield, *Early Schools and School-Books of New England*, pp. 179–182.

__5__. *Reports on the Course of Instruction in Yale College*, pp. 32–33, 53.

__6__. Pike, *A New and Complete System of Arithmetic*, p. 473; Simons, *Bibliography of Early American Textbooks on Algebra*, p. 8.

__7__. Simons, *Bibliography of Early American Textbooks on Algebra*, p. 2.

__8__. Simons, *Bibliography of Early American Textbooks on Algebra*, p. 3. In 1819, Jeremiah Day published a revised, mildly abbreviated edition of his 1814 algebra text addressed to colleges (*An Introduction to Algebra Being the First Part in a Course in Mathematics: Adapted to the Method of Instruction in the American Colleges*) to now include high schools and academies (*An Introduction to Algebra Being the First Part in a Course in Mathematics: Adapted to the Method of Instruction in the Higher Schools and Academies in the United States*).

__9__. Ravitch, *Left Back*, pp. 48, 49; Angus and Mirel, *The Failed Promise of the American High School, 1890–1995*, p. 7.

__10__. National Education Association of the United States, *Session of the Year 1891*, pp. 620–631, 829–830; Mackenzie, “The Report of the Committee of Ten”; Angus and Mirel, *The Failed Promise of the American High School, 1890–1995*, pp. 6–7, 10.

__11__. Ravitch, *Left Back*, p. 41.

__12__. National Education Association of the United States, *Report of the Committee on Secondary School Studies*, pp. 3, 4; National Education Association of the United States, *Session of the Year 1892*, pp. 31, 754.

__13__. National Education Association of the United States, *Report of the Committee on Secondary School Studies*, p. 5.

__14__. National Education Association of the United States, *Report of the Committee on Secondary School Studies*, p. 11.

__15__. National Education Association of the United States, *Report of the Committee on Secondary School Studies*, pp. 7, 8.

__16__. National Education Association of the United States, *Report of the Committee on Secondary School Studies*, pp. 7, 8.

__17__. National Education Association of the United States, *Report of the Committee on Secondary School Studies*, pp. 11–12.

__18__. Ravitch, *Left Back*, pp. 48–50; Angus and Mirel, *The Failed Promise of the American High School, 1890–1995*, pp. 8–17.

__19__. Reeve, “Attacks on Mathematics and How to Meet Them”; Bestor, *Educational Wastelands*; Cairns, “Mathematics and the Educational Octopus”; Mirel and Angus, “High Standards for All?”; Angus and Mirel, *The Failed Promise of the American High School, 1890–1995*, pp. 167–176.

__20__. “The Letter of Admiral Nimitz,” *The American Mathematical Monthly*; Schorling, “Has There Been a Pearl Harbor in Public Education?”

__21__. “Harvard’s Elective System,” *The Harvard Crimson*; Eliot, “Shortening and Enriching the Grammar-School Course.”

__22__. Cajori, *The Teaching and History of Mathematics in the United States*, p. 100.

__23__. Committee of Ten member James C. Mackenzie, headmaster of the Lawrenceville School in New Jersey, hailed from Scotland and was from a modest background. He received almost no formal education prior to his 18th year. Another committee member, Oscar D. Robinson, was a civil war veteran and a nontraditional student in that, while attending a private academy, he completed his studies there at the age of 23 in July 1862. After this, instead of going directly to college, he entered the Union Army as a private in the Ninth New Hampshire volunteer infantry, reportedly seeing action in the major engagements at Antietam, Fredericksburg, Vicksburg, Cold Harbor, and Petersburg. Of the 12 to 15 schoolmates that joined the Ninth with him, only two were left to be discharged from service at war’s end. He entered Dartmouth College afterward and received his degree in 1869 at age 30. Mackenzie, “James C. Mackenzie Papers”; Robinson, “Dr. Oscar D. Robinson.”

__24__. National Education Association of the United States, *Report of the Committee on Secondary School Studies*, p. 51.

__25__. National Education Association of the United States, *Report of the Committee on Secondary School Studies*, pp. 17, 51; Mackenzie, “The Report of the Committee of Ten.”

__26__. It shouldn’t be forgotten that segregation was still firmly entrenched in certain regions of America at this time, being further bolstered by the *Plessy v. Ferguson* (1896) and *Cummings v. Richmond Board of Education* (1899) Supreme Court rulings (and all that these entailed) yet to occur a little over two and five years, respectively, after the release of the Committee of Ten report.

__27__. Harris, “High School—Report of Principal.”

__28__. Hall, *Adolescence*, p. 510.

__29__. Ravitch, *Left Back*, pp. 43–44.

__30__. National Education Association of the United States, *Report of the Committee on Secondary School Studies*, pp. 105, 107.

__31__. National Education Association of the United States, *Report of the Committee on Secondary School Studies*, p. 105. It is worth noting that, though not a member of the Committee of Ten or any of the nine subcommittees, NEA chairman Norman A. Calkins wrote an entire book on the idea of using concrete examples in instruction: Calkins, *Primary Object Lessons for Training the Senses and Developing the Faculties of Children*.

__32__. Simons, *Bibliography of Early American Textbooks on Algebra*; King, *An Analysis of Early Algebra Textbooks Used in the American Secondary Schools Before 1900*, pp. 176–177. It is worth noting that Angie Turner King, on the faculty at West Virginia State College, was highly influential in the education of the mathematician Katherine Johnson of NASA fame. Donoghue, “Algebra and Geometry Textbooks in Twentieth-Century America.”

__33__. National Education Association of the United States, *Report of the Committee on Secondary School Studies*, p. 107.

__34__. Brown, Obourn, and Kluttz, *Offerings and Enrollments in Science and Mathematics in Public High Schools*, p. 29.

__35__. Baker and Jones, *Encyclopedia of Bilingualism and Bilingual Education*, p. 698.

__36__. Poincaré, *Science and Method*, p. 34.

__37__. Thurston, “Mathematical Education.”

__38__. Whitehead, *The Aims of Education and Other Essays*, pp. 70–71.

**CHAPTER 8: ALGEBRA UNCLOAKED**

__1__. The conclusions of the problem work for whole number values of *x* greater than 1 through 7, but for such values, of course, the “days of the week” interpretation loses its meaning.

__2__. *Encyclopedia Britannica* defines an analog computer as “any of a class of devices in which continuously variable physical quantities such as electrical potential, fluid pressure, or mechanical motion are represented in a way analogous to the corresponding quantities in the problem to be solved.” This setup of treating water levels to calculate course averages would qualify as an approximate type of analog computation under this definition. Gregersen, “Analog Computer.”

__3__. It will give the exact value if the final exam averages for each class section were not rounded off (that is, if they were exactly 96, 88, and 54, respectively). It will give a relatively closer value (than the non-weighted average of 79) if any of the three class averages were rounded off (for example, 96.4 rounded off to 96 for Section 001, 87.8 rounded off to 88 for Section 002, and so on). Try it for yourself by making up a sample of 20 final exam scores for a course with 20 students spread over three sections, where Section 001 has 5 students, Section 002 has 11 students, and Section 003 has 4 students, corresponding to the respective proportions of 25%, 55%, and 20% (as was the case when there were 100 students).

__4__. Descartes, *La Géométrie*, pp. 9, 33, 34, 52; Day, *An Introduction to Algebra (Colleges)*, p. 12.

__5__. Plofker, *Mathematics in India*, p. 193.

__6__. Plofker, *Mathematics in India*, pp. 193, 196.

__7__. MLB.com, “Slugging Percentage.”

__8__. Late in 2020, Major League Baseball made the decision to begin including the statistics from the Negro Leagues from 1920 to 1948. While research is still ongoing, the highest slugging percentage in a season will probably end up being either Josh Gibson’s 1937 season (0.974) or Mule Suttles 1926 season (0.877). Baseball Reference, “Single-Season Leaders & Records for Slugging %.”

__9__. Ortega y Gasset, *The Dehumanization of Art*, p. 33.

__10__. If we had five numbers represented by *x*, *y*, *z*, *u*, and *v*, then we would calculate the traditional average by adding them up and dividing by 5 like so: However, with a bit of maneuvering, we can rewrite this as

Because corresponds to 0.20, we could rewrite the rightmost expression as 0.20*x* + 0.20*y* + 0.20*z* + 0.20*u* + 0.20*v*. In quantitative cocktail language, this can be interpreted as five categories represented by *x*, *y*, *z*, *u*, and *v* each contributing 20% to the final result.

__11__. Morse and Brooks, “How U.S. News Calculated the 2021 Best Colleges Rankings.”

__12__. The inclusion of the rankings here is in no way an endorsement of college rankings in general. The author is fully aware of the challenges associated with such ranking algorithms and the diversity of opinion on this highly influential issue.

**CHAPTER 9: ALGEBRAIC FLIGHTS: MECHANISM AND CLASSIFICATION**

__1__. This is illustrated even in the case where the speed is unknown at first. If we know the distance and time of travel, we can use the formula *d* = *st* to solve for the speed *s*. Once we know the value of *s*, however, we can insert it into the formula and generally treat it as a constant from then on for other problems involving that specific scenario.

__2__. Colonel Tina Hartley and V. Frederick Rickey, of the United States Military Academy’s Mathematical Sciences Department, stated in a 2011 presentation that the earliest example they could find of *m* being used as the slope is from the book by John Hymers: *A Treatise on Conic Sections and the Application of Algebra to Geometry* (1837). Hartley and Rickey, “Why Do We Use ‘m’ for Slope?”

__3__. “Radioactivity,” *Merriam-Webster Dictionary*.

__4__. Making estimates such as this one will only yield exactly correct values for the simplest variations, often called linear variations. The situation involving radioactive half-lives varies to a different tune called exponential variation, and linear interpolation can only give an approximate answer.

__5__. As another example, consider *t* = 5 years. Replacing *t* by 5 in the formula yields

This matches the amount given in the table for 5 years. For *t* = 0, we have , which equals one. This is not an intuitive result, but it can be demonstrated algebraically why the zeroth power should take on this value. Though not discussed here, the reasoning can be found in many textbooks on algebra.

__6__. Consider *t* = 3 years for the formula . This yields

This matches the predicted value in the table.

__7__. Check that when *t* = 15 years for the formula , we obtain

This matches the predicted value in the table.

__8__. Splinter, *Illustrated Encyclopedia of Applied and Engineering Physics*, p. 201.

__9__. Splinter, *Illustrated Encyclopedia of Applied and Engineering Physics*, p. 201.

__10__. Here is the calculation of the midpoint value for the expanded IM-5730 table to estimate the time for a sample to get to 1.64 ounces: For the original table with six entries, 1.64 ounces is between 1 ounce and 2 ounces. The halfway point between the years for each of these values in the table is

This value differs from the correct value by 1225 years (14325 – 13100 = 1225).

For the expanded table with 12 entries, 1.64 ounces is now between 1.5 ounces and 2 ounces. The halfway point between the years for each of these values is

This value differs from the correct value by approximately 451 years (13100 – 12649.1 = 450.9).

__11__. Though scholasticism got a bad rap from many scientists, philosophers, and mathematicians of the early scientific era, it represented a revolution in its own right from thinking in earlier medieval periods. Some of the biggest achievements produced by its practitioners included the grand effort to place theology on a foundation of logic (primarily as developed by Aristotle) and reason along with the development and systematic use of much more efficient ways to organize and synthesize large, diverse bodies of information as an aid in that endeavor. Some of these techniques were to prove exceedingly fruitful when adapted to a different platform, that of experimentation and mathematics. Well-known scholastics included St. Anselm, Peter Abelard, Peter Lombard, St. Albertus Magnus, Roger Bacon, St. Thomas Aquinas, and William of Ockham. Longwell, “The Significance of Scholasticism.”

**CHAPTER 10: ALGEBRAIC FLIGHTS: INDETERMINACY AND CURIOSITY**

__1__. O’Connor and Robertson, “Diophantus of Alexandria.”

__2__. Schappacher, *Diophantus of Alexandria*.

__3__. Heath, *Diophantus of Alexandria*, p. 124.

__4__. The way Diophantine equations are usually introduced requires that the coefficients or parameters be integers. This is straightforward to do here by simply multiplying both sides of the equation 0.20*x* + 0.60*y* + 0.20*z* = 80 by 10. Remember, if we multiply both sides of an equation by the same nonzero number, we change the form of the equation but not the solutions. Doing so here unveils the equation 2*x* + 6*y* + 2*z* = 800 (which transforms the coefficients 0.20, 0.60, and 0.20 to the new coefficients 2, 6, and 2 and turns 80 into 800). This new equation will have the same integer solutions as the course average equation, as long as we stipulate that the blended average of the three scores (homework, tests, and final exam) must add to exactly 80 and not be rounded off to 80.

__5__. For a demonstration of why positive integer solutions exist only for *x* + *y* + *z* = 12 (months), see Hayes et al., “Linear Diophantine Equations.”

__6__. The number 1 is the lone exception because it is equal to itself squared.

__7__. The base of the exponent can be thought of as being the generator of the perfect square; that is, the number 7 can be thought of as the generator of 49 because 7 × 7 generates 49, or 13 can be thought of as the generator of 169 because 13 × 13 = 169. Medieval mathematicians often called this generator the *root*, and through their viewpoint naturally extended the name *root* to include values that satisfied any of their second-order equations. Here, we can call 7 a root of the perfect square 49 (or 13 a root of the perfect square 169).

However, 7 can serve as the generator of other values as well. For instance, if we multiply 7 three times, we get 343 (7 × 7 × 7 = 343), or if we multiply 7 four times, we get 2401 (7 × 7 × 7 × 7 = 2401). This means that 7 can also be seen to be a root of 343 and 2401. Using the same term can get confusing, as the way that 7 generates 49 is different from the way it generates 343, which in turn is different from the way it generates 2401, and so on.

To distinguish the various types of generations of a number such as 7, mathematicians give names to the various types of roots: square root (for roots that generate values from being multiplied twice, or squared), cube root (for roots that generate values from being multiplied three times, or cubed), and fourth root (for roots that generate values from being multiplied four times). Thus, 7 is the square root of 49 because it generates 49 by being multiplied twice (49 = 7 × 7 or 7^{2}). Similarly, 7 is the cube or third root of 343 because it generates 343 by being multiplied three times (343 = 7 × 7 × 7 or 7^{3}), and 7 is the fourth root of 2401 because it generates 2401 by being multiplied four times (2401 = 7 × 7 × 7 × 7 or 7^{4}). In a similar fashion, 13 is the square root of 169, the cube root of 2197, and the fourth root of 28,561.

It is important to note, however, that in describing the solutions of a general polynomial equation, the generic term *root* is still in use.

__8__. The quantities represented by *m* and *n* are also sometimes called parameters by mathematicians, but they are not parameters in the exact sense that we have used the term in this book (as scenario variables—fixed within a scenario alongside other quantities that are varying within the scenario).

Sometimes varying quantities, such as the three sides of numerous types of right triangles, can be constructed from other more hidden quantities. But the exact nature of these hidden relationships is sometimes not immediately obvious from their visibly expressed relationships in the form of the Pythagorean Theorem (*x*^{2} + *y*^{2} = *z*^{2}). When mathematicians unearth these hidden relationships to the other quantities, they often introduce a new set of variables to illustrate and represent the connections. This new set of quantities is often also called a *set of parameters*. This is what has happened here with using *m* and *n*.

Clearly, the sides of the right triangles, represented by *x*, *y*, and *z*, are related to each other through the Pythagorean Theorem (*x*^{2} + *y*^{2} = *z*^{2}). But they are also interrelated in an interesting way to another set of quantities—represented by *m* and *n*—that allows us to easily construct integer-valued sides. The splitting of the three unknowns *x*, *y*, and *z* into the respective relationships *m*^{2} – *n*^{2}, 2*mn*, and *m*^{2} + *n*^{2} gives form to this construction through allowing us to simply choose positive integer values for *m* and *n*. These chosen integer values can then, in turn, automatically yield positive integer results for the sides *x*, *y*, and *z*—thus giving us an abundance of the positive integer sides that we sought (as long as *m* is larger than *n*).

__9__. The Babylonians used a base 60 system, called sexagesimal, to represent numbers that, though equivalent in value to our base 10 values such as 45 and 75, have a dramatically different look on Plimpton 322. Neugebauer and Sachs, *Mathematical Cuneiform Texts*, Plate 25.

See a brief discussion of sexagesimal numerals in my earlier book: Williams, *How Math Works*, pp. 212–213.

__10__. Neugebauer and Sachs, *Mathematical Cuneiform Texts*, pp. 38–41.

__11__. Eleanor Robson, “Words and Pictures: New Light on Plimpton 322.”

__12__. Mathematical Association of America, *Paul R. Halmos–Lester R. Ford Awards*.

__13__. Mansfield and Wildberger, “Plimpton 322 Is Babylonian Exact Sexagesimal Trigonometry”; Moss, “Was Geometry Invented by Bureaucrats and Not a Greek Genius?”; Engelking, “Can This Ancient Babylonian Tablet Improve Modern Math?”; Lamb, “Don’t Fall for Babylonian Trigonometry Hype.”

__14__. Høyrup, “Pythagorean ‘Rule’ and ‘Theorem.’”

__15__. Clapham and Nicholson, *The Concise Oxford Dictionary of Mathematics*, p. 170.

__16__. Bell, *Men of Mathematics*, p. 419.

__17__. Matiyasevich, *Hilbert’s Tenth Problem*, pp. 2, 4.

__18__. François Viète, *The Analytic Art*, pp. 67–68.

**CHAPTER 11: A KALEIDOSCOPE OF INGREDIENTS**

__1__. David Pimm, *Symbols and Meanings in School Mathematics*, pp. 88, 122.

__2__. Roscoe, *The Life and Experiences of Sir Henry Roscoe*, p. 81.

__3__. Sagan, *Cosmos*, p. 93.

__4__. MIT Spectroscopy, “Nobel Prizes.”

__5__. Butcher, *Tour of the Electromagnetic Spectrum*, p. 2.

__6__. Chomsky, *For Reasons of State*, p. 402.

__7__. Other important properties that are often affiliated with closure, such as associativity, inverses, and identities, were not pointed out here but are present in real numbers and by consequence some were leveraged in the algebraic processes used.

**CHAPTER 12: GRAND CONFLUENCES**

__1__. *Reports on the Course of Instruction in Yale College*, p. 14.

__2__. Yale University, “Yale University. University Catalogue, 1828.”

__3__. Winterer, *The Culture of Classicism*, pp. 32–34, 60.

__4__. *Reports on the Course of Instruction in Yale College*, pp. 11, 14–15.

__5__. Pereltsvaig, *Languages of the World*, p. 11.

__6__. Descartes, *La Géométrie*, pp. 9, 33, 34, 52.

__7__. Fontenelle, *Eloges des academiciens de l’Academie royale des sciences*, Preface.

__8__. Calvino actually said, “The word connects the visible trace with the invisible thing, the absent thing, the thing that is desired or feared like a frail emergency bridge flung over an abyss.” Calvino, *Six Memos for the Next Millennium*, p. 77.

__9__. Eisner, *The Arts and the Creation of Mind*, p. 11.

__10__. Thurston, Foreword to *Teichmuller Theory and Applications to Geometry, Topology, and Dynamics*.

__11__. Willingham, “What the NY Times Doesn’t Know About Math Instruction.”

__12__. Dewey, *Art as Experience*, pp. 36–39.