Hidden Harmonies

by Ellen Kaplan

  17. Friberg, Remarkable Collection, 337, YBC 6967.

  18. Robson, “Neither Sherlock Holmes”, 167–206.

  19. Meno 82b– 85b.

  20. See, for example, Høyrup, “Mesopotamian Mathematics”, 10.

  21. VAT 6598 and BM 96957. See Robson, “Three Old Babylonian Methods”, 51–72.

  22. Ibid., 63, 64.

  23. Friberg, Amazing Traces, 395.

  24. P. Bergh, in Zeitschrift für Math. u. Physik xxx. Hist-Litt. Abt., 135, cited in Heath’s Euclid I.400–401.

  25. Heath, A History of Greek Mathematics I.61–63.

  26. Robson, “Three Old Babylonian Methods”, 66.

  27. This is also the value reported by Ptolemy for in his Syntaxis I.10, ed. Heiberg I.32.10–35.16.

  CHAPTER 3

  1. While people speak loosely of a tradition ascribing this proof to Pythagoras, in fact the earliest record we have of it dates from about the second century B.C. in China—see our discussion of it in Chapter Five.

  2. See his Euclid I.354–55 and his Greek Mathematics I.149.

  3. Burkert, 110–12.

  4. Aristotle, cited in G. S. Kirk, J. E. Raven, and M. Schofield, 228.

  5. Burkert, 140 n. 109.

  6. Pythagoras’s previous incarnations: ibid., 138, 140 n. 110.

  7. Acusmata listed in ibid., 166–92.

  8. Ibid., 179.

  9. Ibid., 115–17.

  10. Dodgson, A New Theory of Parallels (London: Macmillan, 1888), 16.

  11. Burkert, 156.

  12. Ibid., 142; Kirk, Raven, and Schofield, 228.

  13. Biting the serpent, stroking the eagle, and advent of the white bear: Burkert, 142–43.

  14. Ibid., 433.

  15. Bertrand Russell, Autobiography (New York: Simon & Schuster, 1929), 3:330.

  16. Cited in Barry Mazur’s “How Did Theaetetus Prove His Theorem?” in The Envisioned Life: Essays in Honor of Eva Brann, ed. P. Kalkavage and E. Salem (Philadelphia: Paul Dry Books, 2007), 227.

  17. Music of the spheres, and hearing it from birth: Aristotle, De Caelo B9, 290b12.

  18. Philolaus, from fragment 6, Stobaeus Arith. I.21.7d, quoted in Kirk, Raven, and Schofield, 327.

  19. This non-Pythagorean view is associated with Aristoxenus. See Burkert, 369 ff.

  20. On whether the inner and outer Pythagorean crises coincided, see ibid., 207.

  21. The view that the philosophy of Parmenides exerted a decisive influence on Pythagorean and Greek mathematical thought is forcefully put by Szabó, 248–57 and passim. See too Burkert, 425.

  22. The octave, for example, could not be divided in half. The question of whether this discovery led to or followed from its mathematical analogue is much debated. Burkert, 370 n. 4, thinks it unlikely; Szabó, 173–74, 199 ff., highly probable.

  CHAPTER 4

  1. It is called on directly or indirectly in the proofs of the last six propositions of Book II, five more in Book III (14, 15, 35, 36, 37), and three in Book IV (10, 11, 12).

  2. Schopenhauer’s names for I.47 are quoted in Heath’s Euclid I.354.

  3. The etymology is Heath’s (in his Euclid I.418) by way of Skeat.

  4. Quoted in ibid., I.350.

  5. Euclid’s three lost books of Porisms, ibid., I.10 ff.

  6. Our thanks to our friend and Internet virtuoso, Jon Tannenhauser, for retrieving the magazine’s cover from the World Wide Web.

  7. Heath, Euclid I.417–18.

  8. Quoted by Richard K. Guy in “The Light house Theorem”, Mathematical Association of America Monthly, February 2007, 124.

  CHAPTER 5

  1. George MacDonald Fraser, Quartered Safe Out Here (London: Harvill Press, 1992), 150.

  2. We encountered this proof in Barry Mazur, “Plus symétrique que la sphère”, Pour la Science no. 41, October–December 2003, “Mathématiques de la sphère”, 4.

  3. See Karine Chemla, “Geometrical Figures and Generality in Ancient China and Beyond: Liu Hui and Zhao Shuang, Plato and Thabit ibn Qurra”, Science in Context 18, no. 1 (2005): 159.

  4. We are indebted for our information about Emma Coolidge Weston not only to Jan Seymour-Ford but to Jean Jaworski, executive administrator of the New Hampshire Association for the Blind.

  5. All quotations in this paragraph are from Allyn Jackson, “The World of Blind Mathematicians”, Notices of the American Mathematical Society 59, no. 10, November 2002, 1246–51.

  6. The earlier Bhaskara lived around A.D. 629 in Kerala, the later—often called Bhaskara the teacher—not only five centuries but twelve hundred miles away, at the astronomical observatory in Ujjain.

  7. This material is from Kim Plofker’s “Mathematics in India”, in Katz, 402–3, 411, 477.

  8. Plato, Meno 77A.

  9. Wagner, 71–73.

  10. D. G. Rogers, “Putting Pythagoras in the Frame”, Mathematics Today 44 (June 2008): 123–25. Rogers speaks of his reconstruction as “only a matter of mathematical play, without suggestion that this has any historical basis.”

  11. Chemla, op. cit., 127 n. 11.

  12. This well-known diagram is reproduced from ibid., 149. We follow Chemla’s interpretation.

  13. Quoted in Paul Arthur Schilpp, Albert Einstein: Philosopher-Scientist (New York: Tudor), 1951, 342.

  14. Chemla, op. cit., 141.

  15. Elsewhere he certainly spared no pains to prove (by inscribed and circumscribed polygons of up to 192 sides) that π isn’t 3 but somewhat less than 3.142704, and he says: “Yet a tradition [that π = 3] has been passed down from generation to generation and no one cares to check it. So many scholars followed the tradition that their error has persisted. It is hard to accept without a convincing demonstration” (which he then gives). Of course this correcting (as he says, of fellow scholars, rather than of the venerable sages) amounts to redefining a value through repeated calculation—very different in spirit from contemplating the role and authority of the 3-4-5 right triangle as exemplar rather than example. The passage from Liu Hui is from Joseph W. Dauben, “Chinese Mathematics”, in Victor Katz (op. cit.), 235, and Liu Hui’s demonstration is on 236–37 there.

  16. We have seen this proof variously attributed to Frank Burk (College Mathematics Journal 27, no. 5 [November 1996]: 409) and, on http://www.cut-the-knot.org/pythagoras, to Geoffrey Margrave of Lucent Technologies (#41 on this site), with one variation by “James F.” and another referred to G. D. Birkhoff and R. Beatley, Basic Geometry (a 2000 republication of a text that first appeared in 1959), 92.

  17. Michael Hardy, “Pythagoras Made Difficult”, Mathematical Intelligencer 10, no. 3 (1988): 31. See also http://www.cut-the-knot.org/pythagoras, proof #40.

  18. See Rüdiger Thiele, “Hilbert’s Twenty-Fourth Problem”, Mathematical Association of America Monthly, January 2003, 1–24.

  19. “Proof of the Theorem of Pythagoras”, Alvin Knoer [sic], Mathematics Teacher 18, no. 8 (December 1925): 496–97.

  20. P. W. Bridgman, Dimensional Analysis (New Haven: Yale University Press, 1922). Thanks to Jene Golovchenko in Harvard’s physics department, we’ve been able to trace this application of dimensional analysis to the Pythagorean Theorem back to A. B. Migdal’s Qualitative Methods in Quantum Theory (1977), although G. Polya’s discussion of his 1954 Induction and Analogy in Mathematics could be seen as anticipating it. Perhaps its origin lies farther back in time.

  21. This proof of Tadashi Tokeida’s is from his “Mechanical Ideas in Geometry”, American Mathematical Monthly, October 1998, 697–703. His love of children’s books will make him sympathize with our vicarious Dorothy. See Scott Brodie’s comments on this proof at http://www.cut-the-knot.org, “‘Extra-geometric’ Proofs of the Pythagorean Theorem”. See too four other physics-inspired proofs in Mark Levi’s The Mathematical Mechanic (Princeton: Princeton University Press, 2009): two via the equilibrium of a sliding ring, attached by springs to the ends of its track; another pair involving the kinetics of skating, with a variation picturing two equal hurled masses flying
apart at right angles to their trajectory. Levi notes the contrast between the rigor of the mathematical proof and the conceptual character of the physical.

  22. We found this tiling of the torus and its proof in Ian Stewart’s “Squaring the Square”, Scientific American, July 1997, 96.

  CHAPTER 6

  1. Since Apollonius would have known the law of cosines, Stewart’s proof is occasionally backdated to him by fervent Pergaphiles.

  2. Dijkstra’s generalization of the Pythagorean Theorem is in his manuscript EDW975, http://www.cs.utexas.edu/users/EWD/transcriptions/EWD09xx/EWD975.html. His fantasy about Mathematics Inc. is described in the Wikipedia article on him, “Edsger Wybe Dijkstra”. The varying opinions of him, and contradictory anecdotes, appear in (among others) Krzystof Apt, “Edsger Wybe Dijkstra: A Portrait of a Genius”, http://homepages.cwi.nl/~apt/ps/dijkstra.pdf); Mario Szegedy, “In Memoriam Edsger Wybe Dijkstra (1930–2002)”, http://www.cs.rutgers.edu/~szegedy/dijkstra.html; and J. Strother Moore, “Opening Remarks. . . .”, www.cs.utexas.edu/users/EWD/memorial/moore.html. Nano-Dijkstras are referred to in Alan Kay’s 1997 address to OOPSLA, on YouTube.

  3. Leonhard Euler, Introduction to Analysis of the Infinite, Book I, Chapter VIII, 132.

  4. Most of the biographical material on Johannes Faulhaber is from Matthias Ehmann, “Pythagoras und Kein Ende”, http://did.mat.uni-bayreuth.de/~matthias/geometrieids/pythagoras/html/node11.html. Descartes’s ideas about the orthogonal tetrahedron in four dimensions are in his Cogitationes Privatae X, 246–48. For Tinseau, see http://www-history.mcs.st-andrews.ac.uk/Biographies/Tinseau.html. The fullest life of de Gua that we found is on http://fr.wikipedia.org/wiki/Jean-Paul_de_Gua_de_ Malves.

  5. This proof is due to Jean-P. Quadrat, Jean B. Laserre, and Jean-B. Hiriart-Urruty, “Pythagoras’ Theorem for Areas”, American Mathematical Monthly, June– July 2001, 549–51. An alternative proof is in Melvin Fitting, “Pythagoras’ Theorem for Areas—Revisited”, http://comet.lehman.cuny.edu/fitting/bookspapers/pythagoras/pythagoras.pdf.

  CHAPTER 7

  1. The method of diaresis, or division, as exemplied in Plato’s dialogue The Sophist.

  2. Herbert Spencer Lecture, “On the Methods of Theoretical Physics”, Oxford University, June 10, 1933.

  3. The long but beautiful story behind this insight is well explained in Joseph Silverman’s A Friendly Introduction to Number Theory (Upper Saddle River, N.J.: Prentice Hall, 1997), 169.

  CHAPTER 8

  1. “Et tout ce que l’Idylle a de plus enfantin.” Baudelaire, Paysage.

  2. Henry Reed, “Judging Distances”, which first appeared in New Statesman and Nation 25, no. 628 (March 6, 1943): 155. Published in Collected Poems, ed. John Stallworthy (Oxford University Press, 1991).

  3. This proof is modified from Klaus Lagally’s, as reported in D. J. Newman, “Another cheerful fact about the square of the hypotenuse”, Mathematical Intelligencer 15, no. 2 (1993): 58.

  4. From George Starbuck’s poem “The Unhurried Traveler in Boston”.

  5. And we, the wise men and poets Custodians of truths and of secrets Will bear off our torches and knowledge To catacombs, caverns and deserts. The mathematician L. A. Lyusternik, former member of Lusitania, quoted in Graham and Kantor, 101.

  6. Quoted in Abe Shenitzer and John Stillwell, eds., Mathematical Evolutions. (Washington: the Mathematical Association of America, 2002), 44–45.

  7. Robert Browning, “By the Fire-Side.”

  CHAPTER 9

  1. From John Aubrey’s Brief Lives II, 220–21.

  2. These include John Q. Jordan and John M. O’Malley Jr., “An Implication of the Pythagorean Theorem”, Mathematics Magazine 43, no. 4 (September 1970), 186–89; Jingcheng Tong, “The Pythagorean theorem and the Euclidean parallel postulate”, International Journal of Mathematical Education in Science and Technology 32 (2001): 305–8; and David E. Dobbs, “A single instance of the Pythagorean theorem implies the parallel postulate”, International Journal of Mathematical Education in Science and Technology 33, no. 4 (July 2002): 596–600.

  3. This sequence of lemmas and theorems is from Scott E. Brodie, “The Pythagorean Theorem Is Equivalent to the Parallel Postulate”, http://www.cut-the-knot.org/triangle/pythpar/PTimpliesPP.shtml.

  4. See, for example, what Plato has Simmias say in the Phaedo, 86B.

  5. Our thanks to Jim Tanton for working out this elegant proof.

  CHAPTER 10

  1. You will find much of what we do know in Noam Elkies’s and Nicholas Rogers’s “Elliptic Curves x3 + y3 = k of High Rank”, in Algorithmic Number Theory 3076 (Berlin: Springer, 2004): 184–93; Elkies, “Tables of fundamental solutions (x, y, z) of x3 + y3 = pz3 with p a prime congruent to 4 mod 9 and less than 5000 or congruent to 7 mod 9 and less than 10000”, http://www.math.harvard.edu/~elkies/sel_p.html; and Elkies, “Tables of fundamental solutions (x, y, z) of x3 + y3 = p2z3 with p a prime congruent to 4 mod 9 and less than 1000 or congruent to 7 mod 9 and less than 666”, http://www.math.harvard.edu~elkies/sel_p2.html.

  AFTERWORD

  1. Burkert, 109.

  Footnotes

  a This idea may well have stemmed from the Pythagoreans too. The Greek historian of mathematics Eudemus—not much younger than Euclid—speaking through the later writer Proclus, tells us that constructing a figure with a certain area on a given straight line, called “the application of area”, was “one of the discoveries of the Muse of the Pythagoreans”. Might Euclid have taken even more of this proof from the Pythagoreans? For it is associated with applications of areas in Plutarch’s Symposium: “This [application of areas] is unquestionably more subtle and more scientific than the theorem which demonstrated that the square on the hypotenuse is equal to the squares on the sides about the right angle.” Quoted in Heath’s Euclid, I.343–44.

  b Is this the origin of what is by now a common mathematical trope? You will find it everywhere after, as in the epsilon-delta shims that square up calculus.

  c The intermediate status of Euclid’s constructions—these beings that occupy a middle ground between Being and Becoming, allowing mind to do the moving—is shown in several ways. The miniature Greek drama of each of his propositions begins with a general statement, which is then repeated in a ghostly lettered diagram that, being unscaled, is generic, and made following instructions for ideal rather than real instruments; and always in the chorus, an echo of proofs by contradiction that can’t have diagrams. The drama ends with the return of the general statement, risen from its ghost.

  d We attribute these proofs throughout to Euclid, when in fact he may have no more than anthologized the work of many others—who live, nameless, through him.

  e Inside the front cover of this copy is a printed label: “This book was stolen from Harvard College Library. It was later recovered. The thief was sentenced to two years at hard labor. 1932.”

  f Squaring an infinite series means (we’re sorry to say) multiplying all of its terms by each of its terms, and keeping track of what happens. This would ruin the health of mathematical accountants as surely as recording credit and debit wore out Dickensian clerks—were it not for glimmers of pattern and hopes of order along the way: some terms will negate others, and some will dwindle to nothing as the number of terms increases. Here, when the two squared series are ultimately added together, only the 1 is left standing—as you may surmise by taking just the first two terms of each series, squaring them and adding:

  g What n are we talking about? The n which is the side-length of T, so it is covertly there in the letter T, whose area is n 2.

  h This is legitimate, since n is always greater than 0.

  i Isn’t this once again the battle between taking what counts as belonging to Being or to Becoming, which we’ve seen fought out over mathematics and its objects?

  j Too little given to tell? Bhaskara’s problem supposes that wall and roaming ground are segments of intersecting chords of a circle, the latter a diameter; that the rat is slaughtered at the circle’
s center; and that the student knows that the products of segments of intersecting chords are equal. Hence in this figure, .

  k It was important as well that this was a right triangle: for the Chinese seem to have thought of other triangles as more or less filled with area, but of a right triangle as a framework whose base and height determined its area (thus serving as the foundation for surveying the heavens and earth).

  l This is, after all, an understandable direction that figures might take, from storing, supporting, and enhancing structural arguments to replacing them.

  m Plotinus somewhere sets up a proportion: the immortal soul is to the body as the truth of a theorem is to the proof cobbled together for establishing it.

  n Or was Hilbert right in thinking (like Leibniz before and Einstein after him) that the maximum of simplicity and perfection is realized in the universe? See Thiele, op. cit., 18.

  o Not only Milwaukee’s historian but the offices of VISIT Milwaukee, and of Public Relations, and a local journalist, came up empty-handed. It took the inspired research of Amanda and Dean Serenevy to find articles in obscure journals and relevant years of The Scroll, the Washington High School yearbook—where Alvin is described as “Tall, straight, and thoroughly a man, a fine example of an American.”

  p Generalizations flow through an endless array of nested cones. This theorem, for example, follows easily in a remote inversive geometry, where the ingenuity Ptolemy came up with is quite differently stored in the initial idea of reflections in a circular mirror (which turns his cyclic quadrilateral into line segments). For a proof done in this way, see Tristan Needham’s Visual Complex Analysis (Oxford, 1998).

  q A proof by integration that the areas of similar figures are to one another as the squares of their corresponding linear measures is in the appendix to this chapter.

  r Then as now the spirit of the diagonal hovered helpfully over our mathematical affairs.

  s The shift from working with areas of rectangles to functions of angles is part of the almost glacial movement we have detected, time and again, from a static to a dynamic view of mathematics. While it is true that Euclid speaks in these theorems of choosing “a random point” (such as D), the figure, as it were, then follows this choice and so remains at rest, while looking at it in terms of angles and ratios animates it. As ratios came to take on the status of numbers, Being and Becoming—the figure and ground of content and context—definitively exchanged places.