Physics Textbook Writing: Medieval, Monastic Mimicry

Craig Bohren

Abstract

Textbooks taken in the round are repositories of errors faithfully transmitted from generation to generation. For example, erroneous statements about the speed of light and supposed limitations on the refractive index and its dependence on mass density are pervasive. The history of science in textbooks often bears little resemblance to actual history. The more elementary the textbook, the more scrupulous and knowledgeable its authors must be because most beginning students cannot be expected to know when they are being fed scientific or historical piffle, for which there is no excuse given the several journals devoted to exposition and criticism and the historical resources readily available on the Internet.

The best advice to anyone who would write a physics textbook, especially an introductory textbook, is to adopt the working hypothesis that everything in previous textbooks is wrong. But that is not what usually is done. Like a medieval monk cloistered in a cell decorating illuminated manuscripts but leaving dogma intact, the writer of textbook N dutifully copies what is in textbook N-1, adding a few arabesques but blithely transmitting errors unto the Nth generation. This advice may seem extreme so I’ll soften it a bit by saying that almost every assertion in textbooks in the form of an invariable, unqualified mantra, especially if it asserts supposed limits, is wrong. And the more times the mantra is repeated in print, the more likely it is to be wrong. There are so many examples that it is difficult to know where to begin, but among my favorites are erroneous treatments of refractive indices. I won’t indict any offending textbooks. You can find them for yourselves. What Stephen Jay Gould (1991, Ch.10) calls the "cloning of contents" because "authors of textbooks copy from other texts and often do not read original sources" is not unique to physics, and he gives an amusing example from evolutionary biology. He notes that "good teaching requires fresh thought…rote copying can only indicate boredom and slipshod practice."

The nearly universal textbook statement is that c/n is the "velocity of light" in a medium with refractive index n, which must be greater than 1, the implication being that if it were not Einstein would be dethroned. Well c/n is not the "velocity of light", only one among many, the phase velocity of a plane harmonic wave. Let’s set aside that such a wave cannot exist because it would have to occupy all space and exist for all time. The phase velocity cannot be determined by time-of-flight measurements. It is neither the velocity of a palpable object nor of a signal. Leaf through the three-volume compendium of refractive indices edited by Edward Palik (1998) and you will discover that it is nearly impossible to find a material for which n is not less than 1 at some frequencies. And these are not exotic materials. Try table salt.

When students stumble on refractive indices less than 1, they sometimes are placated with, "Don’t fret. The group velocity can’t be greater than c". They don’t know what the group velocity is, but invoking it makes them go away. Alas, the cure is worse than the ailment because the group velocity not only can be greater than c, it can be negative and less than –c. There are in fact many "velocities of light" (Smith 1972, Bloch, 1977).

Anguish over supposedly nonphysical refractive indices less than 1 was laid to rest more than a century ago by Arnold Sommerfeld. English translations of excerpts from his 1907 paper, his entire 1914 paper, and a 1915 paper by Leon Brillouin, are in Brillouin’s 1960 book. Although the mathematical analysis in these papers, especially Brillouin’s, is formidable, the physical arguments by Sommerfeld, who was reputed to be a superb teacher, can be followed by those innocent of mathematics. He showed that, subject to the restriction of causality—you can’t squeal before you are hurt—no signal can be transmitted faster than c in any medium. Thus for about a century it has been inexcusable for anyone to assert that ⁿ is constrained to be greater than or equal to 1.

This has nothing to do with "negative refractive indices", a better but less newsworthy term for which is "negative phase velocities" (see, e.g., Lakhtakia et al., 2003).

Now let us turn to an even worse textbook botch of refractive indices, the notion that light "slows down" in "denser media". Even Whittaker (1987), in his history of electromagnetic theory, passes this on. On page 11 of Volume 1, in his discussion of Descartes’s corpuscular theory of refraction, Whittaker notes that "These equations imply that…the velocity is greater in the…denser medium. As we shall see, this consequence of the corpuscular theory…is in contradiction with experimental facts." No it is not. There is no necessary relation between mass density (or molecular number density) and phase velocity (c/n). This was pointed out more than 50 years ago by E. Scott Barr (1955) in an outstanding expository paper. His message is succinctly, clearly, forcefully, and humorously conveyed with only one figure and its caption: "Does index of refraction vary directly with density?" This figure shows the refractive index (in the visible) of many liquids versus density, the points connected to answer the question in the caption: NO. But Barr didn’t go far enough because it is easy to find examples that more dramatically demolish fatuous notions about refractive indices (at a given frequency) being greater the greater the density. My favorite example is gold, the refractive index of which in the visible is about one-fifth that of air despite gold being 20,000 times denser.

The non-existence of a universal monotonic relation between refractive index and density can be understood by coming to grips with the concept of refractive index as a phase-shift parameter. It specifies the difference in phase between two plane harmonic waves with the same frequency and propagated the same distance, one in free space, the other in a material medium. The origin of this phase shift is excitation of charges in the medium by an electromagnetic wave. Electromagnetic fields act on charges; masses go along for the ride.

Notions about the proper behavior of refractive indices originated hundreds of years ago when the only light sources were lamps or sunlight, and material samples (e.g., glass, water) were transparent. But generalizing on the basis of a tiny fraction of the electromagnetic spectrum and a restricted class of materials is like pronouncing on the diversity of species solely on the basis of observations made in Kansas.

To criticism that I am unfair because by "denser medium" is meant "optically" denser I have two ripostes. The qualifier "optically" rarely appears, and there is no good reason to redefine refractive index as optical density, especially given the connotations of density and the vagueness of the term optical density [which according to The MacMillan Dictionary of Measurement (1994) is an "imprecise term for transmittance"]. Baptizing refractive index as optical density, and then saying that the (phase) velocity of light is lower in an optically denser medium is logically equivalent to saying that the medium with the higher refractive index has a higher refractive index. True, but not very profound.

Our illustrious predecessors cannot be blamed for arguing about whether light slows down or speeds up in denser media. They didn’t know better. But we do—or should. This is a controversy to be buried along with the ether and phlogiston, not kept alive in textbooks except as a scientific curiosity, a pothole on the road to understanding.

When it comes to the history of science as presented in textbooks, error propagation is rife. Any historian of science can attest that the history of physics in textbooks is mostly, as Henry Ford said, "bunk". To be fair, it often is very difficult to determine who did what first, and hence attribution of laws, constants, theorems, and measurements is almost always wrong. This inspired Stigler’s (1999, Ch. 14) law of eponymy, "No scientific discovery is named after its original discoverer", and Rothman’s (2003, p.xiii) "Infinite Chain of Priority: Somebody Else Always Did it First". For example, who first determined the law of refraction. Harriot, Descartes, Snel? They were latecomers, preceded by around 600 years by Ibn Sahl (Rashed, 1990). Many results of geometrical optics are 17^th century rediscoveries of what was known to Arab scientists 1000 years ago.

Readers may be shocked by my apparent misspelling of Snel. As it happens, this is the correct spelling, easily verified with the Dictionary of Scientific Biography. Why has Snel been misspelled Snell tens of thousands of times? Monastic copyists at work again.

Although it is excusable to get attributions wrong, it is not to pass on what you have not read yourself. For what it is worth, I cannot recall a single instance in which I read in a textbook that scientists said something, only to discover in their original papers that they did not. Again, a few examples will suffice. To judge by textbooks the equations of the electromagnetic field were written down in their present form by Maxwell. Yet if he were to rise from the dead and be presented with his eponymous equations he would not recognize them. They are the work of Oliver Heaviside (see e.g., Nahin, 2002). But a truly egregious example of rewriting history is Newton’s law of cooling in the form, qhΔT, where q is the energy flux, h the “heat transfer coefficient” and ΔT a temperature difference. I have seen books in which not only is this called Newton’s law of cooling, he is cited. So I read the cited reference and discovered that Newton’s law of cooling according to Newton is an exponential decrease of temperature with time of a cooling body, which can be obtained (but was not by Newton) from this supposed law by making several assumptions and approximations (Bohren, 1991). And this equation incorrectly attributed to Newton and called a law is worse than merely historically inaccurate. Unless accompanied by conditions on _h, it is not a law (i.e., verifiable) but rather a definition of h. Because of this and many other experiences, I do not believe any historical statements in textbooks even if accompanied by complete bibliographical information or even direct quotations until I have verified them for myself. I have been deceived too many times.

Given the ready availability of the multi-volume Dictionary of Scientific Biography, contributors to which are not infallible but at least take pride and care in their entries, and the Internet, where one can find many classical scientific papers, there is no excuse for historical bosh in textbooks. Their authors are under no obligation to spice them up with historical tidbits, but if they choose to do so, they have an obligation to get them right. And they also have an obligation to be honest, to note that almost all scientific discoveries were not made by a single scientist. Who gets the credit depends to a large extent on luck, timing, publicity, and nationalism. The electron, for example, had many fathers, despite which J. J. Thomson is lauded as its sole discoverer. Yet “the electron was not discovered by any particular scientist…Several physicists, theoreticians and experimentalists provided evidence that supported the electron hypothesis” (Arbatzis, 2001, p. 188). The norm in science is multiple origins in space and time of discoveries. Moreover, new ideas are not instantaneously accepted because of alleged crucial experiments. But you’d never gather this from the potted histories in textbooks.

There is no excuse for not getting most of the physics right given the many years of publication of journals such as American Journal of Physics, European Journal of Physics, The Physics Teacher, and Journal of Chemical Education. Many papers in these journals are devoted to exposing and criticizing textbook errors. One outstanding example is by Gearhart (1996), who finds that only 6 out of 27 introductory textbooks treat the specific heats of gases and the equipartition theorem correctly. And a remarkably perceptive and thorough criticism of textbook presentations of the photoelectric effect is given by Leadstone (1990), whose sentiments echo my own: “Textbook inadequacies are the rule rather than the exception, and continue to be propagated with remarkable fidelity.” In the same collection a superb essay by French on the role of history in physics teaching includes a figure showing the steady decrease in received frequency of a signal from Sputnik I as it passed overhead, neatly refuting yet another blunder portraying the Doppler effect as an increase followed by a decrease of frequency.

Contrary to what one might think, the more elementary the textbook the more scrupulous and knowledgeable their authors must be. Much time must be spent carefully reading many papers, including (gasp!) original papers of historical importance, weighing each word, being careful not to teach anything that later has to be untaught. Readers of technical monographs can be expected to fend for themselves. But in writing an introductory textbook, authors should keep in mind the words of Thomas Cardinal Wolsey: “Be very, very careful what you put into that head, because you will never ever get it out.” That “head” belonged to Henry VIII, but also likely belongs to most beginning students.

References

Arbatzis, Theordore, 2001: The Zeeman effect and the discovery of the electron, in Histories of the Electron, Jed Z. Buchwald and Andrew Warwick (Eds.), MIT Press.

Barr, E. Scott, 1955: Concerning index of refraction and density. American Journal of Physics, 23, 623-624.

Bloch, S. C., 1977: Eighth velocity of light. American Journal of Physics, 45, 538-549

Bohren, Craig F., 1991: Comment on “Newton’s law of cooling—a critical assessment” by Colm T. O’Sullivan [Am. J. Phys., 58, 956-960 (1990)]. American Journal of Physics, 50, 1044-1046.

Brillouin, Leon, 1960: Wave Propagation and Group Velocity. Academic Press, New York.

French, A. P., 1990: The role of history in physics teaching, in Physicists Look Back: Studies in the History of Physics. John Roche (Ed.), Adam Hilger, Bristol.

Gearhart, Clayton A., 1996: Specific heats and the equipartition law in introductory textbooks. American Journal of Physics, 18, 213-221.

Gould, Stephen Jay, 1991: Bully for Brontosaurus. W. W. Norton, New York.

Lakhtakia, Akhlesh, Martin W. McCall, and Werner S. Weiglhofer, 2003: Negative phase-velocity mediums, in Introduction to Complex Mediums for Optics and Electromagnetics, Werner S. Weiglhofer and Akhlesh Lakhtakia (Eds.), SPIE Press, Bellingham, Washington.

Leadstone, Stuart, 1990: The photoelectric effect—a suitable case for surgery?, in Physicists Look Back: Studies in the History of Physics. John Roche (Ed.), Adam Hilger, Bristol.

Nahin, Paul J., 2002: Oliver Heaviside. Johns Hopkins University Press, Baltimore, Maryland.

Palik, Edward D. (Ed), 1998: Handbook of Optical Constants of Solids. Academic, San Diego.

Rashed, Roshdi, 1990: A pioneer in anaclastics: Ibn Sahl on burning mirrors and lenses. Isis, 81, 464-491.

Rothman, Tony, 2003: Everything’s Relative and Other Fables from Science and Technology, John Wiley & Sons.

Smith, Richard L, 1970: The velocities of light. American Journal of Physics, 38, 978-984.

Stigler, Stephen M., 1999: Statistics on the Table. Harvard University Press.

Whittaker, Edmund, 1987: A History of the Theories of Aether and Electricity: I. The Classical Theories. Tomash/American Institute of Physics.

Craig F. Bohren is Distinguished Professor Emeritus of Meteorology, Pennsylvania State University, the co-author (with Donald Huffman) of Absorption and Scattering of Light by Small Particles, (with Eugene Clothiaux) Fundamentals of Atmospheric Radiation, (with Bruce Albrecht) Atmospheric Thermodynamics, and the author of two popular science books, Clouds in a Glass of Beer and What Light Through Yonder Window Breaks? Email Craig F. Bohren.