2-41. Albert A. Michelson

Born in Strelno, Prussia, Albert A. Michelson (1852–1931) was raised in the United States, attending high school in San Francisco. He obtained a presidential appointment to the United States Naval Academy in 1869, and was commissioned as an ensign in 1873. After a two-year cruise, Michelson was assigned to teach physics and chemistry to midshipmen until 1879, when he joined the office of the Nautical Almanac under Simon Newcomb. The following year, Michelson left Washington to pursue postgraduate studies in Germany, at the University of Berlin, and the University of Heidelberg, as well as in France, at the Collège de France and the École polytechnique. He resigned his commission in 1883 to join the physics faculty of the Case School of Applied Science in Cleveland. Michelson joined Clark University in 1890 in Worcester, Massachusetts, and in 1892, directed the department of physics at the University of Chicago. During World War I, Michelson returned to active duty in the Navy, coming back to Chicago in 1918, where he remained until retirement in 1929.

While at Case, Michelson carried out an experiment in collaboration with Edward W. Morley (1838–1923), a professor at nearby Western Reserve University, to test the isotropy of light propagation. They found that, contrary to all optical theories except that of Stokes, optical phenomena are independent of the translatory velocity of the Earth with respect to the ether, up to the second order of approximation in velocity over the speed of light (Michelson & Morley 1887). Along with the isotropy of light propagation in empty space, the experiment demonstrated mastery of physical optics, and firmly established Michelson’s authority in this domain. Twenty years later, for his work in precision optics Michelson was awarded the Nobel prize in physics, becoming the first American to be so honored. Michelson went on to nominate Poincaré for the 1910 Nobel prize in physics (§ 2-62-26).

Michelson’s surviving correspondence with Poincaré concerns a polemical debate over the adequacy of Fourier-series expansions of discontinuous periodic functions at the point of discontinuity. The controversy began with a letter from Michelson to the editor of Nature in the fall of 1898. Michelson’s account of the problem drew criticism from the Cambridge mathematician A.E.H. Love (1898a), who deplored Michelson’s apparent ignorance of elementary mathematics.

According to J. Willard Gibbs, however, Love had missed the point, as Michelson was concerned with the difference between the graph of the function

$$\lim\limits_{n\to\infty}2\sum_{k=1}^{n}(-1)^{k+1}\frac{\sin(kx)}{k}$$ (1)

and what Gibbs called the “limiting form of the curve” (or the limit curve of the graphs) of the functions

$$y=2\sum_{k=1}^{n}(-1)^{k+1}\frac{\sin(kx)}{k}.$$ (2)

Gibbs described the limiting form of (2) as a sawtooth curve with vertical line segments, and noted that this curve differs from the graph of the limit function (1), the vertical segments of which are empty, save one point on the $x$-axis.

Gibbs’ letter led Love (1898b) to admit to a misunderstanding, but he immediately counterattacked with a criticism of Gibbs’ terminology, arguing that the latter’s limiting form corresponds to limits that are not attained. As for Michelson, in a letter published alongside those of Gibbs and Love in the 29 December 1898 issue of Nature, he maintained his original position.

Michelson’s confident stance may have been buttressed by the graphical evidence provided by a powerful harmonic analyzer he built with his colleague Samuel Wesley Stratton. Michelson seems to have first encountered the phenomenon in the course of developing his harmonic analyzer, although he did not raise the analyzer results as evidence for his point of view.

At this point in time, Michelson could not have been entirely satisfied with the outcome of his exchange with Love. Plausibly, this is what led him to approach Poincaré. As Michelson had hoped, Poincaré took his side, in a letter written the last week of April 1899 (judging by the date of Michelson’s reply). From Paris, Michelson (§ 2-41-2) requested Poincaré’s authorization to communicate his note to Nature, where it appeared on 18 May 1899. While Poincaré judged Michelson to be correct, his letter (§ 2-41-1) did not specify just what Michelson was correct about. Apparently unaware of the Gibbs-Love exchange of the previous December, Poincaré arrived at a conclusion similar to that of Gibbs, whose analysis he effectively extended by identifying the role of the definite integral

$$\int_{0}^{\pi}\frac{\sin x}{x}dx$$ (3)

in determining the graph of (2).

The day after Poincaré’s letter to Michelson appeared in Nature (Poincaré 1899), Love sent another letter to the editor. He capitalized on the ambiguity of Poincaré’s judgment, providing his own interpretation: Poincaré meant only to affirm that the result of summing a series in such a way that $nx$ remains finite while $n$ increases and $x$ decreases to zero is indeterminate. Clearly, Poincaré’s intervention did not bring about the admission of error from Love that Michelson believed he merited. Following the publication of Love’s letter, Michelson encouraged Poincaré to respond, no doubt with a more pointed refutation of Love’s earlier criticism (§ 2-41-3).

Gibbs, however, had given further thought to Michelson’s question. On 27 April 1899, the day before Michelson acknowledged Poincaré’s letter, Gibbs’ second letter to the editor of Nature appeared, responding to Love’s criticism of the previous December. Gibbs observed that his earlier description of the limiting form of (2) was incorrect, but the reason he gave differs from Love’s. He recognized that the vertical segments of the sawtooth extend beyond their points of intersection with the slanted parts, and observed that this overshoot is equal to four times the value of the integral (3). This is what the Harvard mathematician Maxime Bôcher called “Gibbs’ phenomenon.”

Time-stamp: "27.12.2019 19:33"

Notes

• 1 On Michelson’s life, see Livingston (1973); extracts of Michelson’s correspondence have been edited by Reingold (1964). On Michelson’s work see Swenson (1974), Goldberg (1988), Haubold & Pyenson (1988), and Staley (2002).
• 2 Augustus Edward Hough Love (1863–1940) was Sedleian professor of natural philosophy at Oxford from 1899.
• 3 Gibbs (1898); the notation is modified for clarity.
• 4 Michelson 1898. Stratton later directed the National Bureau of Standards, and served as president of M.I.T.; see Kevles (1977, 66, 189).
• 5 See Bôcher (1906); pertinent output of the Michelson-Stratton analyzer is reproduced by E. Hewitt & R. Hewitt (1979, 150).
• 6 Bôcher also offered a proof of Gibbs’ assertion; see Gibbs (1899), and Bôcher (1906). Along with Love, R. B. Hayward (1899) and H. F. Baker (1899) failed to recognize the phenomenon. For further references and details on the history of Gibbs’ phenomenon, see Hewitt & Hewitt (1979). A discussion of the history of representation of discontinuous functions by Fourier series in found in Grattan-Guinness (2005), chapters 25 and 26.