Scientific Method, Statistical Method, and the Speed of Light.

R.J. Mackay and R.W. Oldford ^{∗} Department of Statistics and Actuarial Science University of Waterloo

May 31, 2000

Abstract

What is “statistical method”? Is it the same as “scientific method”? This paper answers the first question by specifying the elements and procedures common to all statistical investigations and organizing these into a single structure. This structure is illustrated by careful examination of the first scientific study carried out by A.A. Michelson in 1879. Our answer to the second question is negative. To understand this a history on the speed of light up to the time of Michelson’s study is presented. The larger history and the details of a single study allow us to place the method of statistics within the larger context of science.

1 Introduction.

“The unity of science consists alone in its method, not in its material.”

Karl Pearson, 1892

page 12 of [?], his emphasis.

“Statistics is the branch of scientific method which deals with the data obtained by counting or measuring the properties of populations of natural phenomena. In this definition ‘natural phenomena’ includes all the happenings of the external world, whether human or not.” M.G. Kendall, 1943 page 2 of [?].

The view that statistics entails the quantitative expression of scientific method has been around since its birth as a discipline. Yet statisticians have typically shied away from articulating the relationship between statistics and scientific method. Perhaps with very good reason. For centuries great minds have debated what constitutes science and its method without resolution (e.g. see [?]). And in this century historical examinations of scientific episodes (e.g. [?]) have cast doubt on method in scientific discovery. One radical position, established by examination of the work of Galileo, is that of the philosopher Paul Feyerabend who writes of method in science:

“… the events, procedures and results that constitute the sciences have no common structure; there are no elements that occur in every scientific investigation but are missing elsewhere.” Paul Feyerabend, 1988 page 1 [?], his emphasis.

Feyerabend then proposed, somewhat facetiously, that the only universal method to be found in science is “anything goes.” Whether Feyerabend’s view holds for science in general is debatable; that it does not hold for statistics is the primary thesis of this paper.

By examining in some detail one particular scientific study, namely A.A. Michelson’s 1879 determination of the speed of light [?], we illustrate what we consider to be the common scientific structure of statistics. There are several reasons for this choice.

First, physical science is sometimes regarded as presenting a greater challenge to the explication of statistical method than, say, medical or social science where populations of individuals are obvious. An early instance is Edgeworth’s hesitation in 1884 to describe statistics as the “Science of Means in general (including physical observations)”, preferring instead the less “philosophical” compromise that it is the science “of those Means which are presented by social phenomena” ([?]).

∗Research supported by the Natural Sciences and Engineering Research Council of Canada

Second, the speed of light in vacuum is a fundamental constant whose value has become “known”; in 1974, it was defined^{1} to be 299,792.458 km/s. So we are in the extremely rare inferential position of “knowing the answer.”

Third, Michelson wrote up his study at a time when it was possible to publish significant amounts of detail, permitting others insight into the difficulties he faced and the solutions he found.

Fourth, the determination of the speed of light has been (and continues to be) important to science and to technology. Consequently its history is rich enough to provide a backdrop on which large scale questions of the nature of science can be discussed.

Fifth, the determinations are known in the statistical literature, first appearing in Stigler’s paper ([?]) on robust estimates of location. Since then, the data have been incorporated into some important introductory textbooks (e.g. [?]) and will no doubt become more commonplace.

Finally, and most importantly, a historical study has the important characteristic of being based entirely on public material. Information gathered together into a single source is information that can be checked against common sources, that can be improved as new historical material becomes available, and that can be a common test bed for others to use. To these ends, we have tried to present the history without reference to method.

The structure which we propose as defining statistical method is described in Section 4. Scientific method is examined in Section 5 and contrasted with statistical method in Section 6. These discussions require separate contexts of differing detail. A broad historical sweep is necessary to appreciate what can be meant by scientific method. It is provided in Section 2, where we give a history of the determination of the speed of light from antiquity to the late 1800s. The stage thus set, the optics, apparatus, and method of Michelson’s first determinations of the speed of light are described in Section 3. These provide the details necessary for discussion of statistical method. A final section explores what we consider to be important ramifications of our approach.

2 Historical background.

The thought of Aristotle (384-322 BC) dominated western science for nearly two millenia. So powerful is his cosmology that it compels him to declare that “… light is due to the presence of something, but it is not a movement” ([?]446^{b}25 − 447^{a}10). No movement, no speed. And if that were not enough, the argument for finite speed is easily dismissed:

Empedocles (and with him all others who used the same forms of expression) was wrong in speaking of light as ‘travelling’ or being at a given moment between the earth and its envelope, its movement being unobservable to us; that view is contrary both to the clear evidence of argument and to the observed facts; if the distance traversed were short, the movement might have been unobservable, but where the distance is from extreme East to extreme West, the strain upon our powers of belief is too great. Aristotle (384-322 BC) On the Soul: Book II 418^{b}20 − 27 [?]

This view was echoed by many thinkers in western history: Augustine (ca 354-430), John Pecham (ca 1230-1292), Albert the Great (ca 1200-1280), Thomas Aquinas (ca 1225-1274), and Witelo (ca 1230-ca 1275) to name a few. So too, the opposite view was argued by some, notably Ibn Al-Haytham (ca 965-1040) and Roger Bacon (ca 1219-1292). But without empirical demonstration to the contrary, the case for instantaneous perception of the source could always be made. In the absence of data, arguments pro and con were forced to be based on the contemporary theory of light, or on interpretation of the conflicting views of ancient authorities, or on established religious doctrines, or on mathematical arguments that demonstrated the necessity or absurdity of one of the alternatives [?].

1By that time the determinations had so little variability that it was considered known to 1 part in 10^{9}, and the standard metre could not be measured to that great a precision. The second is similarly defined; it is the time taken for 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels of the atom Cesium-133. By defining these two quantities all uncertainty was shifted to the unit of distance, a metre, now defined to be distance travelled by light through a vacuum in 1/299792458 second! See [?].

The debate continued into the beginning of the “scientific revolution” of the seventeenth century.^{2} Such giants as Francis Bacon^{3} (1561-1626), Johannes Kepler (1571-1630), and René Descartes (1596-1650), believed the speed to be infinite.

Descartes, for example, likened the transmission of light to that of pushing on a stiff stick – the instant one end (the source) was pushed the other end (the perception) moved (pp. 258-9 of [?]). The analogy is powerful; there is no perceptible movement anywhere along the stick, no matter how long a stick is used! Descartes strongly held this view; when his colleague and scientific mentor, Issac Beeckman (1588-1637), claimed to have performed an experiment which demonstrated the speed was finite, Descartes dismisses the claim saying that if it were true, then Descartes knows nothing of philosophy and his whole theory would be refuted!^{4} Beeckman and Descartes could not agree on an experiment to resolve the issue.^{5}

Among these giants, Galileo Galilei (1564-1642) stands alone in his disagreement; he writes

Sagredo: ... I cannot believe that the action of light, however pure, can be without motion, and indeed the swiftest. Salviati: But what and how great should we take the speed of light to be? Is it instantaneous perhaps, and momentary? Or does it require time, like other movements? Could we assure ourselves by experiment which it may be? Galileo Galilei (1564-1642) Two New Sciences (1638) page 49 of Stillman Drake’s translation of [?]

In the same book, Galileo proposed a demonstration to determine whether light was instantaneous. It was essentially the same that Beeckman had proposed earlier and drew similar fire from Descartes. In a letter to the great experimental scientist Marin Mersenne (1588-1647), dated 11 October 1638, Descartes gave a scathing review^{6} of Galileo’s book; of the proposed demonstration Descartes wrote “His experiment to know if light is transmitted in an instant is useless, since eclipses of the moon, related so closely to calculations made of them, prove this incomparably better than anything that could be tested on earth.” ^{7} Nevertheless, the demonstration was tried in 1667 by members of the Florentine Academy, but without success [?] – light was either instantaneous or near enough so as to be too fast to measure successfully.

In 1676 the first empirical evidence of a finite speed was presented. The Danish astronomer Ole Römer (1644-1710), while working on something entirely different, gathered data and found a discrepancy which led to the discovery. Interestingly, this important and purely scientific discovery came about while working on what we would today call a very applied problem.

2C.D. Lindberg presents preliminary evidence of the debate in medieval Europe [?].

3Bacon had doubts about the infinite speed when considering the great distances that light must travel from the stars to Earth but found such speed easier to swallow given the already fantastic speeds at which stars must travel in their daily orbit about the Earth! See Aphorism 46 of Book II of the Novum Organum e.g. [?]

4From [?] page 307: “Contra ego, si quae talis mora sensu perciperetur, totam meam Philosophiam funditus eversam fore inquiebam.” A rough translation, due to our classically trained colleague G.W. Bennett, is “On the contrary, I would be worried that my entire Philosophy would be on the point of being completely overturned if any delay of this sort were to be perceived by the senses.”

5It is doubtful that Beeckman’s 1629 experiment [?] was successful. The experiment involved firing a mortar and observing its’ flash in a mirror situated some 1851.85 metres away; the movement of a clock situated at the side of the mortar would measure the time elapsed. With today’s value, the time for the flash to reach the mirror and return would be about \frac{1}{100,000} of a second! Descartes argues that even if Beeckman could detect a delay of \frac{1}{24} of a pulse beat (or about \frac{1}{24} of a second yielding a speed of only around 89 km/s), then it should be possible to detect a delay between the occurrence and perception of a lunar eclipse of about one hour. The flaws in this argument are discussed in detail in [?].

6E.g. “... his fashion of writing in dialogues, where he introduces three persons who do nothing but exalt each of his inventions in turn, greatly assists in [over]pricing his merchandise.” Page 388 of [?]. The substantive criticisms are generally directed at Galileo’s not having identified the causes of the phenomena he investigated. For most scientists at this time, and particularly for Descartes, that is the whole point of science.

7Page 389 of [?]. This appears to be based on the argument he gave to Beeckman as described in note 5.

2.1 Longitude.

One of the great practical problems of that time was the determination of longitude, particularly at sea. This could be done by comparing the local time at sea with the time at a fixed reference point — the prime meridian. If, for example, the local time is determined to be two hours earlier than the time at the prime meridian, the location must be 360 × 2/24 = 30 degrees longitude west of the prime meridian.

These times can be determined astronomically. For example local time zero can be defined to be that time when some star, say Arcturus, is observed to cross the imaginary line of longitude running directly north-south through the local position; the corresponding standard time zero would be that time when the same star crosses the prime meridian. Stars are far enough away from us that these two crossings will occur at different moments of time. Carefully determined tables of prime meridian crossing times of various stars would allow navigators to set their local clock. To determine the difference between the local clock and the standard clock, closer astronomical events like an eclipse or occultation of the moon or a planet can be used. These events are observed at essentially the same moment of time whatever the observer’s location on Earth. So comparison of the local time of the close event with its tabulated standard time would give the time difference necessary to calculate longitude.

In 1609, after hearing Flemish reports of a spyglass constructed from two lenses that would enlarge the image of distant objects, Galileo set about the design and construction of the first astronomically useful telescope.^{8} In March of the next year Galileo reported his discovery of the four principal moons of Jupiter [?]. For the first time, here was an orbital system that was demonstrably not centred about the Earth. Galileo argued that this was compelling evidence against the the Ptolemaic system (all celestial bodies revolve around a fixed Earth) and in favour of the Copernican sun-centred system. His public support of the Copernican system as a true representation of the movement of the planets (as opposed to a convenient calculational model) brought Galileo into conflict with those who would interpret certain Biblical passages literally [?]. Some of these people wielded considerable influence within the Catholic church of Rome; by order of Pope Urban VIII he was banned from further publication and placed under house arrest from 1633 until his death in 1642. This did not prevent him from continuing his scientific work.^{9}

But this momentous scientific discovery also had commercial potential — King Philip III of Spain offered a handsome prize to anyone who could come up with a practical method of determining a ship’s position when out of sight of land. Galileo hit upon the idea of using the predicted times of the eclipses of Jupiter’s moons to provide the common celestial clock necessary to determine longitude. In November of 1616 he began negotiations with Spain for navigational uses of his astronomical discoveries and in 1617 worked on developing a telescope for use at sea while continuing his negotiations with Spain [?]. Unfortunately the tables he produced were not accurate enough for their intended purpose — the theory at the time did not account for the perturbations of the moons due to their mutual interaction [?].

Although many writers advocated their use at sea, those who appreciated the practical difficulty of directing a very long telescope at Jupiter while aboard a lively ship were skeptical and undoubtedly amused by the proposed method. It was never to become successful at sea. But on land, very accurate determinations of longitude could be obtained this way and resulted in a substantial reform of geography in the 17th and 18th centuries.

2.2 The first evidence.

In 1671 Römer went to Hven, an island community near Copenhagen, to help redetermine the longitude of the observatory located there. With others he began observing a series of eclipses of Io, Jupiter’s largest moon. In the end they had eight months of observations or, since Io makes one revolution of Jupiter in 42 hours, timings on about 140 eclipses over 2/3 of the year. The time intervals between these eclipses were not regular but depended on where the Earth was in its orbit. The length of the interval was shorter when the Earth approached Jupiter than it was when the Earth moved away from the planet. The mathematically predicted time of an eclipse was too early if the Earth was near Jupiter and too late if the Earth was far from Jupiter. This systematic lack of fit allowed Römer to announce in Paris in September 1676 that the eclipse predicted for November 9 that year would actually occur 10 minutes later. The observation bore him out and Römer argued that the discrepancy was due to the finite speed of light — the light takes longer to reach us the farther we are from its source.

8According to Stillman Drake ([?] page 29), Hans Lipperhey a lens grinder from the Netherlands is generally assigned credit for the telescope’s invention and applied for its patent in 1608.

9Today’s visitor to Florence’s Museum of Science can find a glass and ivory case displaying an ironic relic – Galileo’s bony middle finger pointing heavenward.

From his observations, Römer estimated that light takes about twenty-two minutes to cross the full diameter of Earth’s orbit or about eleven minutes for light from the sun to reach us on Earth. From this he estimated its speed to be about 214,000 kilometres per second.^{10}

Römer’s “proof” was not immediately accepted by all. Alternative explanations were provided by Gian Domenico Cassini (1625-1712) then also an astronomer at the newly formed Academie des Sciences in Paris. He had observed inequalities in time intervals that depended on the location of Jupiter in its own elliptical orbit. And in 1666 Cassini had published tables on the eclipses of the satellites of Jupiter from which work he also noticed the discrepancy. He had briefly considered a finite speed of light in 1675 but soon rejected it for a more traditional explanation. Cassini, and later his nephew Giacomo Filippo Maraldi (1665-1729), suggested that Jupiter’s orbit and the motion of its satellites might explain the observed inequalities ([?], [?] and [?]). Many astronomers continued to hold the view that light was instantaneous.

It was not until a study by James Bradley (1693-1762)^{11} was reported in 1729 that nearly all agreed that the speed is finite. Bradley had been studying the parallax of the stars and discovered an annual variation in the position of stars that could not be explained by the parallax effect. However, it could be explained by the motion of the Earth if light’s speed were finite. Based on careful observations, he estimated that light took eight minutes and twelve seconds to reach the Earth from the sun resulting in a value for light’s speed of 301,000 km/sec.

In 1809, based on observations on the eclipses of Jupiter’s moons for 150 years, Jean-Baptiste Joseph Delambre (1749-1822) estimated the time taken by light to travel from the sun to Earth to be eight minutes and 13.2 seconds resulting in a speed of about 300,267.64 ≈ 300,300 km/sec.^{12}

The results of these early astronomical estimates are summarized in Table 1.

Year Authors  Observational Source Speed (km/sec)
1676 Römer  Jupiter satellites  214 000
1726 Bradley  Aberration of stars  301 000
1809 Delambre Jupiter satellites  300 300

Table 1: Studies based on astronomical observation.

Unfortunately, measurements of the speed made in this way depended on the astronomical theory and observations used. Simon Newcomb (1835-1909) tells of an inaugural dissertation in 1875 by Glasenapp whereby observations of the eclipses of Io from 1848 to 1870 show that widely ranging values for the speed “could be obtained from different classes of these observations by different hypotheses” ([?] page 114). It was shown that values for the sun to Earth time could be produced between 496 and 501 seconds resulting in speeds between 295,592.8 ≈ 295,600 and 298,572.6 ≈ 298,600 km/s. ^{13}

Better determinations of the speed might be made if both source and observer were terrestrial. Because all would then be accessible, greater control could be exerted over the study and hence the observations. But this brings us back to the age old problem: how could the speed be measured terrestrially?

2.3 Terrestrial determinations.

Imagine two people standing at either end of a very long track. The first uncovers a powerful light source at an appointed time and the second records the time at which the light is seen. The length of the track divided by the difference between the start time and the time the light is perceived would give a measurement of the speed of light.^{14} The trouble, of course, is that light is so fast that the distance must either be very large or the time taken very small. Extremely large distances and extremely short time intervals are very difficult to measure directly.

10For more on Römer see [?]. For more detail on this study see [?].

11See [?] and [?].

12The time here is as reported in [?]. To calculate the speed, the distance between the Earth and sun must be known. In the estimate reported here, the distance used was 148,092,000 km as derived from Bradley’s figures above.

13Again, using Bradley’s Earth to sun distance.

14This is essentially the experiment proposed by Isaac Beeckman to Descartes in 1629. See footnote ??.

Matters can be improved some if both observers have light sources which they cover with a screen. Time measurement begins when the first observer removes the screen sending light to the second. The second light source is uncovered when the second observer sees the first. Now when the first observer sees the second light source he again screens his source. The time between uncovering and covering the first light source is a measure of the time light takes to travel twice the distance between the two observers. The improvements are obvious: the distance is doubled and a single clock has replaced two supposedly synchronized clocks. Here was Galileo’s proposed study of 1638; nearly 200 years would pass before it was improved sufficiently to produce results.

The necessary innovations were introduced by Hippolyte Fizeau (1819-1896). One innovation was to replace the second person by a fixed flat mirror whose surface is perpendicular to the beam of light from the source. If this could be done, then the light beam would be reflected directly back at its origin and so remove completely one human source of variation from the system. The second innovation was to automate the covering and uncovering of the source, thereby further reducing the variation from the first human source. Together, these allowed Fizeau to replace the direct measurement of time with an indirect measurement of speed.

Rather than measure time between uncovering and covering, Fizeau could measure the minimum speed that the screen must travel in order to cover the source at the exact time the light returns. The trick was to use an accurately machined toothed wheel placed spinning in front of the source to act as the moving screen. The teeth would screen the source while the gaps would uncover it and so the wheel acted just as Galileo’s observer. Any light returning to the source would strike either a tooth or a gap thus causing a flashing light to return. The faster the wheel spun, the dimmer this return image would appear. If the wheel was set spinning fast enough that every beam sent out struck a tooth on its way back, no image would be observed. Twice this speed should produce a continuous beam as the beam sent out returned through the next available gap. Three times the speed produces no light, and so on. This speed of rotation, coupled with the distance travelled (twice 8,633 metres in Fizeau’s setup), could be transformed into a measure of the speed of light. In this way, Fizeau produced the first terrestrial determination of the speed of light in 1849.

Others were quick to build on this monumental achievement. Only two years later Leon Foucault (1819-1868), a former collaborator of Fizeau, produced more accurate measurements based on a rotating mirror rather than a toothed wheel.

3 Michelson’s 1879 determinations of the speed of light

In November of 1877 Albert Abraham Michelson (1852-1931), then a twenty-four year old ensign in the US Navy and an instructor in physics at the U.S. Naval Academy in Annapolis Maryland, hit upon the means to improve Foucault’s rotating mirror approach. Even then, he needed to conduct many preliminary studies before being confident of an improved value for the speed of light. In his own words ([?] page 115) “Between this time and March of the following year a number of preliminary experiments were performed in order to familiarize myself with the optical arrangements. Thus far the only apparatus used was such as could be adapted from the apparatus in the laboratory of the Naval Academy.”

In April he initiated contact with Professor Simon Newcomb (1835-1909) of the US Navy ([?] page 38) who was then superintendent of the navy’s Nautical Almanac and renown in the navy and the scientific community as an astronomer. Michelson discussed his work and methods with Newcomb. At this point however, Michelson was still an unknown who would not be funded by the US Navy for such specialized research. Fortunately, having married Margaret McLean Heminway in the spring of 1877, he could turn to a wealthy father-in-law for financial support. His father-in-law^{15} had become deeply interested in Michelson’s preliminary results and in July of 1878 provided him the $2000 necessary to purchase the fine optical instruments to carry out his measurements. So began a lifelong quest for the speed of light.

15Referred to in [?] only as a “private gentleman”.

3.1 Optical theory.

One of the difficulties with having great distances between the source and the mirror is that the intensity of the light will decrease with distance. So as to keep the image as bright as possible, a lens is placed between the source of the light and the mirror. If, as in the diagram below, the source, S, and the mirror, M are

Figure 1: S and M are placed at the point-source focus of each other.

placed so that a point-source light from one is focused precisely on the other, then the return image will be as bright and as crisp as possible.

Note that the distance between L and M is not equal to that between L and S. As M moves farther from the lens, S will need to be moved closer in order for both points to remain at the focus of the other’s point source. This is true provided both points are beyond the focal length of the lens (that point where beams of light parallel on one side of the lens would meet on the other side).

By moving S and M farther apart, all the while keeping each at the other’s point focus, we increase the distance the light must travel and therefore the time it will take. Even so, the time taken is exceedingly short and difficult to measure.

Instead of Fizeau’s wheel, Foucault used a rotating mirror interposed between S and L as in the next diagram.^{16} Light rays from the source that strike R and proceed through the lens L will strike M and return

Figure 2: Interposing a mirror, R, between the source S and the lens L.

to the source S. If after the light beam first strikes R outbound from S, R can be rotated before it is struck again by the beam returning from M, then the returning beam will no longer return exactly to the source S but will instead be deflected away from S in the direction of the rotation. The amount of this deflection is exactly twice the angle that R was rotated.

The solution then is to rotate the mirror fast enough that it changes its orientation before the light strikes it on return from M. By rotating it at a constant speed, the amount of deflection will be the same for all light beams that go through L, strike M and return. Then for a continuous beam of light from S and a constant high speed of rotation of R, an image of the source will appear beside S instead of coincident upon it (as shown in Figure 4). The faster R rotates the farther the returned image, I, will be displaced from the source, S.

By carefully measuring the amount of displacement from S to I (see Figure 4), and the distance from S to R, the tangent of the angle of deflection can be determined as |IS|/|SR|. Together with the fixed speed of rotation, this angle can be used to measure the time it took light to travel the distance from R to M and back. Dividing distance by time gives a measurement of the speed of light.

16According to Newcomb (page 117) this had been suggested much earlier by Charles Wheatstone (1802-1875) and tried without success by Dominique Francois Jean Arago (1786-1853) in 1838.