## 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).

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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