Abstract
Much of the mathematical reasoning employed in the typical introductory physics course can be traced to Pythagorean roots planted over two thousand years ago. Besides obvious examples involving the Pythagorean theorem, I draw attention to standard physics problems and derivations which often unknowingly rely upon the Pythagoreans’ work on proportion, music, geometry, harmony, the golden ratio, and cosmology. Examples are drawn from mechanics, electricity, sound, optics, energy conservation and relativity. An awareness of the primary sources of the mathematical techniques employed in the physics classroom could especially benefit students and educators at schools which encourage integration of their various courses in history, science, philosophy, and the arts.
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Notes
See for example chapters III–V of Heath (1965).
By the term “Greek” I signify the entire empire falling under ancient Hellenistic culture around the Mediterranean.
The signatures on the space and time dimensions take a bit of hand-waving at the introductory level, but can be interpreted with Minkowski’s original convention of using an imaginary unit for the time coordinate in spacetime.
For a quadrilateral ABCD inscribed inside a circle the product of the diagonals equals the sum of the products of the opposite sides, or \(AC \cdot BD = AB \cdot CD + AD \cdot BC. \) Ptolemy stated his theorem in terms of chords on a circle (the analog to our sines), not in the modern language of trigonometric or circular functions, which are after all sophisticated analytic abbreviations encoding the definite ratios attaining between lengths (or chords) defined on circles. Ptolemy motivates the importance of this theorem for his, and indeed all, work on geometry and astronomy by first stating, "But we see that it is first necessary to explain the method of determining chords: we shall demonstrate the whole topic geometrically once and for all” (Ptolemy 1998 [2nd century CE], (I 9), p. 47).
In his Harmonices Mundi (Kepler 1997 [1619], p. 264) Kepler writes a digression on the translation of Greek mathematical terms which concludes “We must therefore keep the custom introduced by the barbarian commentators on the Greek Elements.” This is essentially the terminology carried to this day. At least in mathematical texts the word “proportion” derives from proportio, the common Latin translation for the Greek \(\alpha\nu\alpha \; \lambda \acute{o} \gamma o \nu. \) The word "ratio" translates the Greek \(\lambda \acute{o} \gamma o \varsigma, \) a notoriously potent word which, depending upon ones discipline and context outside science and mathematics, unpacks variously as ratio, reason, proportion, number, meaning, means, cause, wisdom, or word.
This form matches their original definitions which employed words. For example Kepler (Kepler 1997 [1619], p. 166) recalls the ancients’ definition of “harmonic proportion as that in which, three numbers being placed in their natural order, the amounts by which one of the pair of neighbors exceeds the other are in the same proportion as the outer numbers.” The harmonic mean is the middle term of the three numbers (that is, the symbol b in Eq. 12).
For example, the Pythagoreans demonstrated that the musical scale can be constructed theoretically (as opposed to empirically with the human ear) using the three classical means: arithmetic, geometric, and harmonic. Consider their application to a string with length 1 and another string of length 1/2 (octave). It turns out the arithmetic mean between these two extremes is 3:4 (fourth) and the harmonic mean is 2:3 (fifth). Thus the aesthetic foundation of octave, fourth and fifth discussed in previous section "falls out” mathematically. The remainder of the major scale follows by considering the third classical mean, the geometric, which fills in the remaining whole notes on scale (e.g., D is geometric mean between C and E). See (Fideler 1987, pp. 327–328).
From Eq. 15 we have ac = b 2 which is equivalent to stating a/b = b/c.
This form holds for the center of percussion r P relative to the body’s center of mass. See (Hibbeler 2009, p. 508).
Shakespeare alludes to this music: Look how the floor of heaven / Is thick inlaid with patines of bright gold: / There’s not the smallest orb which thou behold’st / But in his motion like an angel sings, (Shakespeare, The Merchant of Venice, act 5, scene 1.)
There is debate and historical confusion on whether this “central fire” was the Sun or a separate “central hearth.” Regardless of which, the Pythagorean model is striking in considering earth as a spherical, mobile body.
The names for the conic sections originate (Heath 1965, p. 150–151) from the Pythagorean technique in geometrical algebra known as “application of areas.” It is essentially a geometric way to solve quadratic equations by finding an equivalent area to a given figure, one easier to reckon than the given figure. In more general use this application (παραβ o λη) could be exceeding \((\nu\pi\epsilon\rho\beta o \lambda\eta)\) or \(falling\;short\) \((\epsilon\lambda\lambda\epsilon\iota\psi\iota\varsigma)\) of the given figure. I include the Greek to show the origin of our terms parabola, hyperbola and ellipse. Indeed these terms were adopted by Apollonius (3rd century BCE) in his famous treatise on conics since his constructions employed the Pythagorean application of areas, where the three possible applications (equal, exceeding or falling short) lead to the most general construction of the three conics.
Although Kepler’s first two laws can already be seen in his New Astronomy (see Kepler (1992 [1609])) he introduces his third law (p 2 ∝ a 3) in The Harmony of the World. The entire contents of the final Book V (On the Most Perfect Harmony of the Heavenly Motions) lays out the consequences of this third “harmonic law.” There he derives several musical scores for the planetary motions (e.g., see Kepler 1997 [1619], p. 457) accompanied by demonstrations such as “Proposition XI: The proportion of the motion of Saturn at aphelion to that at perihelion ought to have been 4:5, a major third, but that of Jupiter’s motions 5:6, a minor third.” At the risk of digression we will not unravel the question of what extent this music of the spheres could or should actually be heard. The most creative answer to this question is related by Aristotle who writes that “the sound is in our ears from the very moment of birth and is thus indistinguishable from its contrary silence” (Aristotle 1941b [4th century BCE], Book II, Ch. 9). Porphyry holds that these are sounds “which we cannot hear because of the limitations of our weak nature” (Porphyry 1987 [3rd century CE], p. 129). For a modern treatment of this question using astrophysical examples see Caleon and Ramanathan (2008).
References
Allie S., Buffler A., Campbell B., Lubben F., Evangelinos D., Psillos D., Valassiades O. (2003). Teaching measurement in the introductory physics laboratory. The Physics Teacher, 41(7), 394–401. doi:10.1119/1.1616479
Alonso M., Finn E. (1970). Physics. Addison-Wesley series in physics. Boston: Addison-Wesley.
Archytas. (1987 [4th century BCE]). The fragments of Archytas. Grand Rapids: Phanes Press.
Aristotle. (1941a [4th century BCE]). Metaphysics. In R. P. McKeon (Ed.), W. D. Ross (Trans.). New York: Random House.
Aristotle. (1941b [4th century BCE]). On the Heavens. R. P. McKeon (Ed.) (J. L. Stocks, Trans.). New York: Random House
Aristotle. (1941c [4th century BCE]). Physica. R. P. McKeon (Ed.) (R. P. Hardie & R. K. Gaye, Trans.). New York: Random House
Boyer, C. (1968). A history of mathematics. New York: Wiley
Bunge, M. (2003). Twenty-five centuries of quantum physics: From Pythagoras to us, and from subjectivism to realism. Science & Education, 12, 445–466.
Caleon, I., Ramanathan, S. (2008). From music to physics: The undervalued legacy of Pythagoras. Science & Education, 17, 449–456
Caleon, I. S., & Subramaniam, R. (2007) From Pythagoras to Sauveur: Tracing the history of ideas about the nature of sound. Physics Education, 42(2), 173
Caspar, D., & Klug, A. (1962). Physical principles in the construction of regular viruses. In A. Chovnick (Ed.), Cold spring harbor symposia on quantitative biology (Vol. 27, pp. 1–24). Cold Spring Harbor : Cold Spring Harbor Laboratory Press.
Coldea, R., Tennant, D., Wheeler, E., Wawrzynska, E., Prabhakaran, D., Telling, M., Habicht, K., Smeibidl, P., Kiefer, K. (2010). Quantum criticality in an ising chain: Experimental evidence for emergent E8 symmetry. Science, 327(5962), 177–180
Copernicus, N. (2004 [1543]). On the revolutions of heavenly spheres (De revolutionibus orbium coelestium). In S. Hawking (Ed.). (Charles Glen Wallis, Trans.). Philadelphia: Running Press.
Douady, S., Couder, Y. (1992). Phyllotaxis as a physical self-organized growth process. Physical Review Letters, 68(13), 2098–2101. doi:10.1103/PhysRevLett.68.2098
Fideler, D. (1987). The formation and ratios of the Pythagorean scale. Grand Rapids: Phanes Press.
Finocchiaro, M. (2008). The essential Galileo. Massachusetts: Hackett Pub. Co.
Heath, T. (1956). The thirteen books of Euclid’s elements. Dover classics of science and mathematics. Mineola: Dover Publications.
Heath, T. (1965). A history of Greek mathematics: From Thales to Euclid. No. v. 1 in Oxford Books. Oxford: Oxford Clarendon Press
Heath, T. (1991). Greek astronomy. Dover books on astronomy. Mineola: Dover Publications.
Hecht, E. (2009) Einstein on mass and energy. American Journal of Physics, 77(9), 799–806. doi:10.1119/1.3160671
Heyrovska, R. (2005). The Golden ratio, ionic and atomic radii and bond lengths. Molecular Physics, 103(6), 877–882
Hibbeler, R. (2009) Engineering mechanics: Dynamics (12th ed.). Upper Saddle River: Prentice Hall
Hoggatt, V. Jr., Bicknell-Johnson, M. (1979). Reflections across two and three glass plates. The Fibonacci Quarterly, 17, 118–142
Holton, G., Elkana, Y. (1997). Albert Einstein: Historical and cultural perspectives. Dover Science Books. Mineola: Dover Publications
Iamblichus. (2003 [3rd century BCE]). The life of Pythagoras (T. Taylor, Trans.). Whitefish: Kessinger Publishing, LLC.
Kepler, J. (1981 [1596]). The secret of the universe (Mysterium Cosmographicum) (E. J. Aiton, Trans.). Janus series. Norwalk: Abaris Books.
Kepler, J. (1992 [1609]). New astronomy (Astronomia Nova) (W.H. Donahue, & Gingerich, O., Trans.). Cambridge: Cambridge University Press
Kepler J (1997 [1619]). The harmony of the world (Harmonice Mundi). In E. J. Aiton, A. M. Duncan, & J. V. Field (Eds.) Memoirs of the American Philosophical Society. Philadelphia: American Philosophical Society
Lavatelli, L. (1964). The Pythagorean numbers in vector problems. American Journal of Physics, 32(11), 850–852. doi:10.1119/1.1969915
Levi, M. (2012). The mathematical mechanic: Using physical reasoning to solve problems. Princeton: Princeton University Press.
Levine, D., & Steinhardt, P. J. (1984). Quasicrystals: A new class of ordered structures. Physics Review Letters, 53(26), 2477–2480. doi:10.1103/PhysRevLett.53.2477
Livio, M. (2002). The golden ratio. New York city: Broadway Books.
Neugebauer, O. (1969). The exact sciences in antiquity. Dover classics of science and mathematics. Mineola: Dover Publications
Newton, I. (1995 [1687]). The principia (A. Motte, Trans.). Great minds series. Amherst: Prometheus Books.
Pais, A. (2005). Subtle is the lord: The science and the life of Albert Einstein. Oxford: Oxford University Press.
Papastavridis, J. (1998). Tensor calculus and analytical dynamics. Library of engineering mathematics, Taylor & Francis.
Porphyry. (1987 [3rd century CE]). Vita Pythagorae. Grand Rapids: Phanes Press.
Ptolemy, (1980 [2nd century CE]). Die Harmonielehre des Klaudios Ptolemaios. Original text in Greek; editor’s commentary in German by I. Düring, New York: Garland Publishing
Ptolemy, (1998 [2nd century CE]). Ptolemy’s almagest (J. G. Toomer, Trans.). Princeton: Princeton University Press.
Srinivasan, T. P. (1992). Fibonacci sequence, golden ratio, and a network of resistors. American Journal of Physics, 60(5), 461–462. doi:10.1119/1.16849
Styer, D. (2000). The strange world of quantum mechanics. Cambridge: Cambridge University Press.
Tipler, P. (1999). Physics for scientist and engineers: Mechanics, oscillations and waves, thermodynamics (4th ed., Vol. 1). New York: W.H. Freeman
Uhden, O., Karam, R., Pietrocola, M., Pospiech, G. (2012). Modelling mathematical reasoning in physics education. Science & Education, 21, 485–506. doi:10.1007/s11191-011-9396-6
Wigner, E. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications in Pure and Applied Mathematics, 13(1), 1–14
Young, H., Freedman, R., Ford, L. (2006). University physics (12th ed.). Addison-Wesley series in physics. Boston: Addison-Wesley
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Clarage, J.B. The Pythagorean Roots of Introductory Physics. Sci & Educ 22, 527–542 (2013). https://doi.org/10.1007/s11191-012-9553-6
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DOI: https://doi.org/10.1007/s11191-012-9553-6
Keywords
- Circular Motion
- Golden Ratio
- Pythagorean Theorem
- Musical Scale
- Similar Triangle