Pierre Fermat's Proof of the Sine Law of Refraction and Purpose

Pierre Fermat's Proof of the Sine Law of Refraction and Purpose

Copyright 1998 by Emerson Thomas McMullen

Abstract

Pierre Fermat used the idea of God's purposes acting efficiently in nature to mathematically prove the sine law of refraction

Descartes and Fermat

Both Rene' Descartes (1596-1650) and Pierre de Fermat (1601-1665) had discovered analytic geometry - so who was the better mathematician? There was no doubt in Descartes' mind that he was superior. In spite of Descartes' opinion, Fermat was the better mathematician.⁽¹⁾ One reason is that he proved the sine law of the refraction of light. He was also the superior physicist because he handled the theoretical aspects of the behavior of light better than Descartes. In accomplishing this, Fermat based his proof of the sine law on the idea of efficient purpose in nature. This paper explores Fermat's use of purpose.

Purpose in science developed from one of Aristotle's four causes, the final cause. Final cause asked "Why?" or "For what purpose?" did something exist. For example, in Aristotle's science, the final cause (purpose) of an acorn is to be an oak tree. The acorn may never achieve this end, but nevertheless that is its purpose. Aristotle's purpose existed in Nature, but scholars in the Middle Ages Christianized this concept, and final causes became God's purposes.

Today, scientists do not consider using purpose in science, (asking "why?"), a proper methodology. Function, (asking "How?"), is what they concentrate on. Function deals with the "how" of a watch: gears and the like. Purpose explains that the "why" of a watch is to tell time. However, during the Scientific Revolution, the use of purpose in science was acceptable practice. Most productive was the assumption of efficient purpose, which was often expressed by the saying "Nature does nothing in vain."⁽²⁾ Scientists often used God and Nature interchangeably when discussing purpose, but in either case, they were referring to intelligent intention in the design of the creation.⁽³⁾ Optics is especially suitable for applying efficient purpose and those investigating reflection and refraction frequently did so.

Kepler

Johannes Kepler (1571-1630) made a major step in the study of optics by rethinking the nature of light. Up to his time researchers tended to think of light as a free agent, having a will and a mind of its own, so to speak. Kepler argued that light was passive and behaved according to geometric laws. This full commitment to the mathematiciation of light was a dramatic break with the past and set the stage for all future studies. Mathematics does not involve the idea of purpose, but in other contexts, including speculation about extraterrestrial life, Kepler used purpose in his reasoning.

Kepler's 1604 Astronomiae pars optica (The Optical Part of Astronomy) formed the foundation for the optical studies that accelerated in the seventeenth century. (Besides Kepler's insightful 1604 book and his Dioptrice (1611), this acceleration occurred particularly because of the inventions of the telescope and microscope.) Kepler's work on optics was especially foundational concerning the refraction of light.⁽⁴⁾ Drawing an analogy about little spheres in motion, Kepler correctly thought of light as slowing down in a denser medium. This went against Alhazen (c. 965-1040) and Witelo of Silesia (1235-1275), both of whom exhibited conflicting ideas on the refraction of light. For instance, in one place Alhazen referred to light's motion as instantaneous, and in another as having its swiftness affected by different media.⁽⁵⁾

Refraction

Refraction is the change in direction of light (or a wave) as it passes from one medium to another. This change is caused by the change in the velocity of light (or the wave). Descartes appears to have drawn his explanation about how light refracted somewhat indiscriminately from the above sources. Since he usually did not reference anyone, no one can really know what came from where. In a rare admission, Descartes wrote in a letter that "Kepler was my first teacher in optics."⁽⁶⁾ Coming from the prideful Descartes, this is high praise.

The modern expression of the sine law of the refraction of light is given by the equation:

n sin r = sin i

where i is the angle of incidence, the angle the incident light ray makes to the normal; r is the angle of refraction, the angle the refracted ray makes with the normal; and n is the velocity of light in the medium of refraction divided by the velocity in the medium of incidence. In the figure, CFG is a light ray refracted at F due to the change from one medium to another at AB. CFH forms the angle of incidence and GFD forms the angle of refraction. Stated in the seventeenth century, the law is that the sine of the angle of incidence is directly proportional to the sine of the angle of refraction (or in the above equation, n is a constant). This relationship may have been first discovered by Thomas Harriot (1560-1621), and then independently by Willebrord von Roijen Snel (1580-1626). Descartes, who knew Snel very well, was the first to publish it. While he correctly described the sine law, strangely, Descartes offered no correct proof. How could he have arrived at the right answer when he had no apparent pathway to it?

Fermat and Descartes

Fermat was a first-class mathematician whose geometrical discoveries such as analytical geometry rested on firm foundations. He is famous for his 'last' theorem, which had challenged mathematicians up to its recent solution.⁽⁷⁾Descartes had some trouble believing these facts even when they became exceedingly clear during his dispute with Fermat. He called Fermat a troublemaker and insinuated to others that Fermat owed his reputation to a few unsystematic lucky guesses.⁽⁸⁾ The fact that Fermat never published his work did not help the situation either. Instead, he followed the standard procedure of his day in reporting his mathematical discoveries like analytical geometry, by letters sent to friends. For example, Fermat and Blaise Pascal (1623-1662) worked out probability theory in a 1654 exchange of letters. Generally publishers did not want to risk time and money in the small mathematics market.

Descartes

A cursory look at Descartes' sine law of refraction prompts one to wonder whether he was the lucky guesser. It is hard to understand how he calculated the correct answer, given his approach. Descartes thought of light propagation as a pressure instantaneously transmitted through a plenum. According to him, our eyes see an object as a blind man 'sees' an obstacle through the pressure of a walking stick held in his hand.

Copying Kepler's little spheres, Descartes explained light refraction by comparing it to the motion of a tennis ball piercing a cloth that represents the interface of two different media. He showed that the sine of the angle of incidence of the ball's trajectory is proportional to the sine of the new (refracted) angle of the ball's trajectory in the second medium. This result is correct. Strangely, in reaching this result, he followed the ideas of Alhazen and required that the speed of light, previously described as instantaneous, now exhibit change. As with Kepler, this change occurs in an instant, and at the density-interface. Then, breaking with Kepler, Descartes conceived that, for a denser medium, the speed of the light increases in proportion to the density. This is the opposite of what would happen with any ball. The ball would be slowed by the denser medium. Descartes' physical analogy with the ball not only breaks down, but is very tricky in the first place since light, in his philosophy, being pressure, does not move at all.⁽⁹⁾

Further, similar to Alhazen and Witelo, Descartes' mathematical treatment separated the trajectory of refracted light into vertical and horizontal components. He made the vertical component change, but wrongly left the horizontal component unchanged.⁽¹⁰⁾ Descartes should have continued to build upon Kepler's foundational thought instead of on an eclectic mix. When subjected to mathematical criticism, Descartes' analogy makes no sense. No wonder historian of science Richard S. Westfall labeled it preposterous.⁽¹¹⁾ Descartes' explanation of refraction was so unconvincing that Gottfried Wilhelm Leibniz (1646-1716) charged that Descartes only succeeded in obtaining the sine law because he already knew the answer from Snel in Holland.⁽¹²⁾ (Leibniz also thought that Snel had succeeded by using purpose.⁽¹³⁾Others who thought that Descartes copied Snel's work were Isaac Vossius, who made the charge public in 1662, and Christiaan Huygens (1629-1695), who had expressed his concerns even earlier.⁽¹⁴⁾

Interestingly, Descartes wrote in a letter to Isaac Beeckman that if his assumption of the instantaneous propagation of light were wrong, then his entire natural philosophy would be wiped out at its very foundation.⁽¹⁵⁾ Descartes had also made a similar statement involving the movement of the heart. He was wrong on the velocity of light and heart motion, but he was right about his philosophy.

Fermat

Understandably, Fermat had not accepted Descartes' geometrical explanation for the sine law of refraction and suspected its validity. "Descartes has proved nothing," he complained in a 1657 letter.⁽¹⁶⁾ Fermat resolved to prove the sine law. There were an infinite number of proportions different from the correct one;⁽¹⁷⁾ it was his hope that Descartes' result might be close, but not identical with his. Fermat may have entertained this hope because he had come to the correct conclusion that, due to resistance, the speed of light is slower in a denser medium. This view was in line with Kepler, but contrary to that of Descartes.⁽¹⁸⁾ It is easy to see why Fermat expected a different answer.

Unlike the wealthy Descartes, who inherited his money, Fermat had to work for a living. He was a lawyer and a jurist and did not have much spare time to pursue his many mathematical interests. Years passed before he was able to prove the sine law of refraction, and by then there was experimental verification of the law. He continued searching for the proof, pressured on by his correspondents. In the end, using his mathematical method of maximums and minimums,⁽¹⁹⁾ he arrived at the same result as Descartes. In the approach that he took, Fermat made a specific assumption which he could not test: he assumed that light travels slower in a denser medium. He combined this assumption with the hypothesis, based on the idea of efficient purpose, that nature always acts by the shortest course.⁽²⁰⁾ By this, he meant that nature follows the easiest course, or uses the least amount of time, which is not necessarily the same as the shortest path. Combining the two provided Fermat a specific theory that he thought he could prove mathematically.

Fermat could test the idea that light does not follow the shortest path between different media. Everyone has observed that an oar, or a straight stick, appears bent when dipped partially into water. Similarly, a well-aimed spear thrust straight at a fish, or an arrow shot at it by a hunter above the water, always misses. Obviously light is not traveling along the shortest route. If it were, the arrow or spear, which is moving in a straight line, would hit the fish. (A hunter above the water has to aim under the fish to hit it.) From these observations, Fermat could conclude that light was following a longer, bent path. Here nature, which does nothing in vain, was not following the shortest path; it was traveling the path that results in the least amount of time. This assumption has come to be known as Fermat's principle.

In the letter in which he announced his proof of the sine law, and an earlier one to the same correspondent, Fermat referred to the principle "that nature always acts by the shortest courses."⁽²¹⁾ This principle works for reflection, where the incident angle equals the reflected one, but not for refraction. For refraction, Fermat reasoned that "nature performs its movements by the simplest courses."⁽²²⁾ Eventually he interpreted the simplest course to be that which took the least time.

Fermat's Principle

Let us briefly examine Fermat's principle of least time as it applies to the refraction of light at a pond. If the speed of light were the same in air as in water, then the shortest path would be a straight line from one given point to another - for example, from your eye to the surface of the water to a fish in the pond. However, the speed of light is faster in air than in water and, following Fermat's principle, the shortest time for the light ray is not along a straight line. To travel from the fish to your eye as fast as possible, the light ray would "want to" get out of the pond quickly because it moves more slowly there. Going to the surface of water by a shorter path presents the ray with a longer total distance to travel, but it "makes up" for the extra distance because it can go faster in the air.

Many sailboat races have been won by the skipper who sailed farther away from the race course and caught a stronger wind before it reached the other boats. The winning boat traveled a longer distance than the others, but did it in a shorter time. This is the case for the refraction of light. The light travels a longer distance than a straight line, but it covers that distance faster than if it had gone along the straight path. Naturally, there is one route that is the fastest. Fermat reasoned that nature would 'send' light along this fastest, but not shortest, path. It took him a while to figure out what that path would be. When he arrived at the optimum result, he was surprised to find those angles to be the same as reported by Descartes.

Fermat was content to allow the priority of discovery to go to Descartes while claiming that he had supplied the proof. Was this proof good science? He combined the guiding idea that nature does nothing in vain with the specific assumption that light travels faster in rarer media. From this he arrived at the testable hypothesis that refracted light travels along the path resulting in the least time. Did it matter that the hypothesis was based on efficient purpose?

Fermat defended his methodology. He found support for his application of purpose to physics by appealing to Galileo Galilei (1564-1642). Fermat pointed out that Galileo had used efficient purpose in connection with his law of naturally accelerated motion.⁽²³⁾ In developing his definition of accelerated motion in Dialogues Concerning the Two New Sciences (1638), Galileo referred to the ". . . custom and procedure of nature herself in all her other works, in the performance of which she habitually employs the first, simplest, and easiest means."⁽²⁴⁾ In later passages, Galileo developed this idea, using the terms "exceedingly simple" and "more simple." Galileo was echoing Nicholas Copernicus (1473-1543) who advised that "we should rather heed the wisdom of Nature. Just as it especially avoids producing anything superfluous or useless, so it frequently prefers to endow a single thing with many effects."⁽²⁵⁾ Isaac Newton (1642-1727) also wrote about efficient purpose, stating that nature is pleased with simplicity and does not affect the pomp of superfluous causes.⁽²⁶⁾

In optics, the idea of efficient purpose has an extensive pedigree. Besides being stated by Galileo, Descartes also used it. In his discussion of the nature of light in Le Monde (1664), Descartes used purpose in reasoning that Aristotle could have stated. Contradicting his stand against the use of purpose in science, Descartes wrote that "while Nature has several ways to arrive at a given effect, she always [and] infallibly follows the shortest."⁽²⁷⁾ Descartes was merely following the long tradition of the use of purpose in optics. In some form or another, this historical line can be traced back through Witelo⁽²⁸⁾, Claudius Ptolemy (2nd century), and even earlier.⁽²⁹⁾ In fact, Descartes used tables of refraction from Witelo, and Witelo had copied these nearly exactly from Ptolemy.⁽³⁰⁾

Huygens

Christiaan Huygens developed his law of refraction from considering light as a wave phenomenon.⁽³¹⁾ Huygens' work supports Fermat's principle and his assumption of a slower velocity of light in a denser medium. Speaking of his theory of refraction, Huygens continued to employ purpose (as he had in the diverse contexts of extraterrestrial life and spontaneous generation). He stated that his theory of refraction "employs the least possible time. . . .. M. Fermat was the first to propound this property of refraction, holding with us, and directly counter to the opinion of M. Descartes, that light passes more slowly through glass and water than through air."⁽³²⁾ Note how convinced Huygens sounds in this passage. He refers to Fermat's purpose-based hypothesis as a "property of refraction." Huygens is sure of this because he arrived at his own proof of the sine law of refraction. While developing his wave theory of light, he produced independent scientific support of Fermat's principle.

Conclusion

Fermat's principle of least time, born out of the idea of efficient purpose, is still in use today. For instance, it is being applied to the theoretical analysis for non-destructive testing procedures.⁽³³⁾ Further, as Ernst Mach (1838-1916) observed, the theoretical interpretation of the law of refraction began with Fermat and Huygens. Descartes and Snel had put forth expressions that are only imitations of refraction by geometrical constructions.⁽³⁴⁾ When Descartes is examined closely, he cannot effectively explain how refraction works, much less prove the sine law. Mach called Descartes' explanation "unintelligible and unscientific."⁽³⁵⁾ Fermat's accomplishment not only makes him a better mathematician than Descartes, but also a better theoretician.

References

1. See Carl B. Boyer, The History of the Calculus and Its Conceptual Development (New York: Dover Publications, Inc., 1949/1950) especially p. 154.

2. This concept is fully presented and developed in my William Harvey and the Use of Purpose in the Scientific Revolution, forthcoming.

3. An example of using God and Nature interchangeably is William Harvey (1578-1657), who referred to ". . . the finger of God or nature . . . ." in "On Generation," The Works of William Harvey, trans. R. Willis (Philadelphia: The University of Pennsylvania Press, 1989) p. 402.

4. J. Kepler, "Ad Vitellionem Paralipomena, quibus Astronomiae Pars Optica traditur....," trans. S.M. Straker in Kepler's Optics: A Study in the Development of Seventeenth-Century Natural Philosophy, Ph.D. Dissertation, Indiana University, 1970, pp. 533-534.

5. Ibid, p. 494.

6. W. R. Shea, The Magic of Numbers and Motion: The Scientific Career of Rene' Descartes (Canton, MA: Science History Publications, 1991) p. 154.

7. Some originally thought that Fermat's theorem had been solved because of heavy publicity to that effect in 1993. For example, see Barry Cipra, "Fermat's Last Theorem Finally Yields," Science, 1993, 261:32-33. However, the proof was not complete and difficulties were encountered in filling in the technical details. This was not publicized as was the initial announcement. I. Peterson, "Fermat Proof Flaw: Fixing the Details," Science News, 1994, 144:406. Finally, the complete proof was announced in 1995. Barry Cipra, "Princeton Mathematician Looks Back on Fermat Proof," Science, 268:1133-4. The theorem is that

xⁿ + yⁿ = zⁿ has no solution for n>2 where x, y, z and n are positive integers.

8. M.S. Mahoney, "Fermat, Pierre de." D.S.D., vol. 4.

9. Bruce S. Eastwood argues Descartes is only trying to make the sine law comprehensible in "Descartes on Refraction: Scientific versus Rhetorical Method," Isis, 1984, 75:481-502.

10. Neil M. Ribe, "Cartesian Optics and the Mastery of Nature," Isis, 1997, 88:51.

11. R. S. Westfall, The Construction of Modern Science (Cambridge: Cambridge University Press, 1977) p. 53.

12. G. W. Leibniz, Discourse on Metaphysics, trans. P. G. Lucas and L. Grint (Manchester: Manchester University Press, 1953) p. 40.

13. Ibid, p. 39.

14. A. I. Sabra, Theories of Light from Descartes to Newton (Cambridge: Cambridge University Press, 1981), pp. 101-102.

15. R. Descartes, Letter to Beeckman, 22 August 1634, Oeuvres de Descartes, Vol. I, eds. C. Adam and P. Tannery (Paris: Liberairie Philosophique J. Vrim, 1959-1967) p. 308.

16. Pierre de Fermat, Oeuvres, Vol. II, ed. Paul Tannery and Charles Henry (Paris: Gauthier-Villors and Sons, 1894) p. 356.

17. Ibid, p. 461.

18. Sabra, Theories of Light, p. 117.

19. See Boyer, The History of the Calculus, pp. 155-9, for an exposition of this method.

20. Fermat, Oeuvres, Vol. II, p. 354. This August, 1657 letter contains his first approach to the problem. The complete solution is found in a January, 1662 letter to the same correspondent. Ibid, p. 457. A modern presentation of Fermat's solution to the problem of refraction is in Chapters Four and Five of Sabra's Theories of Light from Descartes to Newton, and in Appendix I of M.S. Mahoney's The Mathematical Career of Pierre de Fermat (1601-1665) (Princeton: Princeton University Press, 1973/1994).

21. ". . . la nature agit toujours par les voies les plus courtes." Fermat, Oeuvres, Vol. II, pp. 354 and 458.

22. ". . . la nature fait ses mouvements par les voies les plus simples." Ibid, p. 355.

23. Ibid, p. 359; Sabra, Theories of Light, p. 157, note #57.

24. Galileo Galilei, Dialogues Concerning the Two New Sciences, trans. Stillman Drake (Madison: The University of Wisconsin Press, 1974) p. 153. In this passage, Galileo refuses to go beyond the inference to ask why nature employs the simplest means. Later in this same section, Simplicio speaks of how accelerated motion leads to great hidden mysteries related to the creation of the universe and "to the residence of the first cause." Salviati answers that "such profound contemplations belong to doctrines much higher than ours." Ibid, p. 182. Galileo here introduces an idea which will find its fulfillment after Newton. Scientists should be satisfied with finding out "how?" and not feel bound to explain "why?" This attitude helped undermine the use of purpose in post-Newtonian science.

25. Nicholas Copernicus, On the Revolutions, trans. E. Rosen (Baltimore: The Johns Hopkins University Press, 1978/1992) p. 20.

26. Isaac Newton, Principia, trans. Andrew Motte, revised by Florian Cajori (Berkeley: The University of California Press, 1962) p. 398.

27. Oeuvres de Descartes, Vol. 11, eds. C. Adam and P. Tannery (Paris: Leopold Cerf, 1909) p. 89. The translation is by A.M. Smith, Descartes's Theory of Light and Refraction: A Discourse on Method (Philadelphia: The American Philosophical Society, 1987) pp.16-17. This is one of several examples of purpose in Descartes' philosophy. Another is Boyle's classification of Descartes' principle of the conservation of the total quantity of motion as grounded in purpose.

28. Vitellonis, Thuringopoloni, Books II and V in Opticae thesaurus, ed. F. Risner (Basel: 1572) pp. 61 and 192. (A Landmarks in Science microprint.) Also see Witelo, Perspectiva, Book V, trans. A.M. Smith, Studia Copernicana, Vol. 23 (Warsaw: The Polish Academy of Sciences Press, 1983) p. 90.

29. A. M. Smith, "Saving the Appearances of the Appearances: The Foundations of Classical Geometrical Optics," Archive for History of Exact Sciences, 1981, 24:98. Christopher Kaiser argues that thirteenth-century investigators of optics contributed to scientific development by focusing more on structures and laws in nature than on efficient, formal, and final causes. See his Creation and the History of Science (Grand Rapids: William B. Eerdmans Publishing Co., 1991) p. 79.

30. A. M. Smith, Descartes's Theory of Light and Refraction: A Discourse on Method (Philadelphia: The American Philosophical Society, 1987) p. 47, footnote #1.

31. Huygens' method of analyzing light waves inspired Nobel Laureate Richard Feynman. Feynman adapted Huygens' technique to quantum waves and devised a new way of computing wave functions. See Herbert, Quantum Reality: Beyond the New Physics (New York: Doubleday, 1985) p. 115.

32. Christiaan Huygens, Treatise on Light, trans. Silvarnus P. Thompson (New York: Dover Publications, 1962) pp. 42-43.

33. See Chau-Chi Ku, Diffraction of Spherical Sound Pulses by a Circular Cylinder, (Cornell University Ph.D. Dissertation, 1981). Fermat's principle is the starting point in a method for the mapping of crack-line flaws in homogeneous media in Andrew Norman Morris'Ray Methods for Inverse Problems of Elastic Wave Scattering (Northwestern University Ph.D. Dissertation, 1981).

34. Ernst Mach, The Principles of Physical Optics, trans. John S. Anderson and A.F.A. Young (London: Dover Publications, 1926) pp. 39-40.

35. Sabra, Theories of Light, p. 104.

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Acknowledgments

I thank T. Africa, C. Briggs, C. Crouch, G. Hickey, S. Peacock, K. Skidmore-Hess and J. Woods.