The Origins of the Velocity-Distance Relation

Struve O. and Zebergs V. , Astronomy of the twentieth century (New York, 1962), 469.

Whitrow G. J. , “Hubble, Edwin Powell”, Dictionary of scientific biography, vi, 528–33 (p. 531). On Hubble see also Mayall N. U. , “Edwin Powell Hubble”, Biographical memoirs of the National Academy of Sciences, xli (1970), 175–214.

Sciama D. W. , Modern cosmology (Cambridge, 1971), 44.

Huggins W. , “Further observations on the spectra of some of the stars and nebulae, with an attempt to determine therefrom whether these bodies are moving towards or from the Earth; also observations on the spectra of the Sun and of Comet II, 1868”, Philosophical transactions, clviii (1868), 529–64.

Keeler J. E. , “Spectroscopic observations of nebulae”, Lick Observatory publications, iii (1894), 161–229. After Keeler's studies of nebulae around 1900, and until the 1920s, the nebulae that were not of a diffuse or planetary type were usually classed together as ‘spirals’. The spirals were to become the candidate external galaxies in the 1910s, and by the middle of the 1920s it had become generally agreed that the spirals were indeed galaxies.

Campbell W. W. , “Some peculiarities in the motions of the stars”, Lick Observatory bulletin, vi (1911), 125–35. The judgment that the older the star, the faster its speed, relied on the then-accepted theory of stellar evolution. This theory's central thesis was that as a star ages, it changes its colour from white to red and alters its spectral type along the sequence O, B, A, F, G, K, M, R, N: See Smith R. W. , “Russell and stellar evolution”, Dudley Observatory report, no. 13 (1977), 9–13.

Slipher V. to Lowell P. , 2 January 1913 (Lowell Observatory Archives).

Slipher's measurements were published in “The radial velocity of the Andromeda nebula”, Lowell Observatory bulletin , no. 58 (1913). See also Hetherington N. S. , “The measurement of radial velocities of spiral nebulae”, Isis, lxii (1971), 309–13.

For an account of Percival Lowell's astronomical investigations and the early years of the Lowell Observatory see Hoyt W. G. , Lowell and Mars (Tucson, 1976).

Slipher V. to Lowell P. , 3 December 1910 (Lowell).

Miller J. to Slipher V. , 13 June 1913, quoted by Hall J. in “Vesto Melvin Slipher, 1875–1969”, Yearbook of the American Philosophical Society (1970), 161–6 (p. 164).

Campbell W. to Slipher V. , 9 April 1913 (Lowell).

Wolf M. to Slipher V. , 13 June 1914 (Lowell).

Slipher V. to Wright W. , September 1914 (Lowell).

Slipher V. , “Spectrographic observations of nebulae”, Popular astronomy, xxiii (1915), 21–24.

Campbell W. to Slipher V. , 2 November 1914 (Lowell).

For an explanation of this method see, for example, Smart W. M. , Text-book on spherical astronomy (Cambridge, 1962; fifth edition), 263–75.

Truman O. H. , “The motions of the spiral nebulae”, Popular astronomy, xxiv (1916), 111–12.

Young R. K. and Harper W. E. , “The solar motion as determined from the radial velocities of spiral nebulae”, Journal of the Royal Astronomical Society of Canada, x (1916), 134–5.

Hertzsprung E. to Eddington A. S. , 18 May 1916 (Hertzsprung papers, Aarhus University).

Eddington A. S. to Hertzsprung E. , 13 June 1916 (Aarhus).

Slipher V. to Campbell W. , 18 December 1916 (Lick Observatory Archives).

Slipher V. , “Nebulae”, Proceedings of the American Philosophical Society, lvi (1917), 403–9.

Slipher , op. cit. (ref. 15).

Paddock G. F. , “The relation of the system of stars to the spiral nebulae”, Publications of the Astronomical Society of the Pacific, xxviii (1916), 109–15.

Some writers have claimed that C. A. Wirtz was the first to introduce the K term into these studies. This error seems to have been made first by Hubble in his The realm of the nebulae (London, 1936), 107. J. D. North followed Hubble's account in his The measure of the universe (Oxford, 1965), 142. North's volume is the best available history of theoretical cosmology in the first half of the twentieth century. However, North's interpretations of a number of the observational developments that affected cosmology in this period have been challenged by recent research which has made extensive use of archive sources: See, in particular, Berendzen R. Hart R. and Seeley D. , Man discovers the galaxies (New York, 1976) and Smith R. W. , The history of the island universe theory 1900–1931 (unpublished Ph.D thesis, University of Cambridge, 1978).

Campbell W. , Stellar motions. With special reference to motions determined by means of the spectrograph (New Haven, 1913), 208.

Wirtz C. A. , “Über die Bewegungen der Nebelflecke”, Astronomische Nachrichten, ccvi (1918), 109–16.

See also Fernie J. D. , “The historical quest for the nature of the spiral nebulae”, Publications of the Astronomical Society of the Pacific, lxxxii (1970), 1189–230 (p. 1221).

Lundmark K. , “The relations of the globular clusters and spiral nebulae to the stellar system. An attempt to estimate their parallaxes”, Küngl. Svenska Vetensamps Akademiens Handlingar, Band 60, no. 8 (1920), 24.

Wirtz C. A. , “Einiges zur Statistik der Radialbewegungen von Spiralnebeln und Kugelsternhaufen”, Astronomische Nachrichten, ccxv (1921), 349–54.

Einstein A. , “Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie”, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften (1917), 142–52. This is printed in translation in The principle of relativity (Dover edition, 1952), 177–88. Our notation for the field equations follows North , op. cit. (ref. 26), 72–81.

Einstein's use of ‘curved’ space to explain the properties of the universe was not novel: See North , op. cit. (ref. 26), 72–81.

Quoted in Moszkowski A. , Einstein the searcher: His work explained from dialogues (London, 1921; translation by Brose H. L. ), 127.

In 1929 Jeans asked the rhetorical question: “Are there any limits to the extent of space?” Even a “generation ago, I think most scientists would have answered in the negative. They would have argued that space could be limited only by the presence of something which is not space” ( Jeans J. H. , The universe around us (Cambridge, 1929), 70). General Relativity, he proclaimed, had removed this objection and had laid open the way to the acceptance of a finite universe.

Einstein , op. cit. (ref. 32), 188.

Einstein A. , “Spielen Gravitationsfelder im Aufber der materiellen Elementarteilchen eine wesentliche Rolle?”, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften (1919), 349–56. This is printed in translation in The principle of relativity (ref. 32), 191–8.

Mach E. , The science of mechanics (Chicago, 1902), 229–38. This is a translation by McCormack T. J. of the second edition of Mach's Die Mechanik in ihrer Entwickelung historisch-kritisch dargestellt.

Blackmore J. T. writes that Einstein had “couched the essential insight of his theory of general relativity, the equivalence of inertial and gravitational mass, within a number of Machian ideas. Indeed, he considered the theory as a natural consequence of Mach's earlier work and believed that except for accidental factors Mach quite likely would have discovered and elaborated the theory himself” ( Blackmore J. T. , Ernst Mach: His life, work, and influence (London, 1972), 254). On 25 June 1913 Einstein wrote to Mach to tell him that from the “basic and fundamental assumption of the equivalence of the acceleration of the reference frame and of the gravitational field” it is a necessary consequence that “inertia has its origin in a kind of mutual interaction of bodies” (quoted in Blackmore , op. cit., 254).

Eddington A. to de Sitter W. , 11 June 1916 (Leiden Observatory Archives) and Eddington A. to de Sitter W. , 4 July 1916 (Leiden). In his June letter Eddington had commented: “Hitherto I had only heard vague rumours of Einstein's new work [on the field equations]. I do not think anyone in England knows the details of this paper.”

See Kahn F. and Kahn C. , “Letters from Einstein to de Sitter on the nature of the Universe”, Nature, cclvii (1975), 451–4.

de Sitter W. , “On Einstein's theory of gravitation, and its astronomical consequences. Third paper”, Monthly notices of the Royal Astronomical Society, lxxvii (1917), 3–28.

Whitrow G. J. Professor pointed out to me that the de Sitter model was the first that contained a time-horizon (or event-horizon in current terminology). That is, there is some distance, say R, from which an observer at the origin could never have any information of events occuring at R or beyond. Hence, one can speak of a horizon to the universe at R. Now a different observer with a different origin would place his horizon differently, and so the spatially-closed nature of the model applies to a particular observer. See also Tolman R. C. , Relativity, thermodynamics and cosmology (Oxford, 1934), 354.

Kahn and Kahn , op. cit. (ref. 41), 453.

Einstein , op. cit. (ref. 32), 183.

De Sitter , op. cit. (ref. 42), 25. In the same paper he used a distance of 50,000 parsecs as the characteristic separation of the galaxies.

De Sitter , op. cit. (ref. 42), 24.

de Sitter W. , “On the possibility of statistical equilibrium of the universe”, Proceedings of the Royal Academy of Sciences of Amsterdam, xxii (1922), 866–8. He wrote that there was as yet no physical criterion with which to decide between Solutions A and B (p. 868).

Douglas Vibert A. , The life of Arthur Stanley Eddington (London, 1956), 39–43. At the island of Principe in the Gulf of Guinea the weather was poor and so the Sobral results were of greater weight. Typical of the excitement caused by the findings was Shapley's remark to a correspondent: “What do you think of the principle of relativity now that the extreme case of the Einstein theory has been proved by the English eclipse results?” ( Shapley H. to Benioff H. , 7 December 1919 (Shapley papers, Harvard University Archives)).

Hale G. E. to Dyson F. W. , 9 February 1920, quoted by Clark R. W. in Einstein: The life and times (London, 1973), 238.

Douglas , op. cit. (ref. 49), 44.

Eddington A. S. , The mathematical theory of relativity (Cambridge, 1923), 161. Typical of this increased interest was the letter Hertzsprung sent to the British astronomer Reynolds J. H. in May 1923. Hertzsprung was excited because the “circumstance, that the only negative ‘radial velocities’ found for spiral nebulae belong to the two largest… just invites [us] to examine still fainter objects of this kind. I am pretty convinced, that the spiral nebulae have really great velocities by themselves. The question is for me, if there is superposed a de Sitter-effect in the displacements observed of the spectral lines” ( Hertzsprung E. to Reynolds J. H. , 2 May 1923 (Aarhus)).

Eddington A. to de Sitter W. , 16 August 1917 (Leiden).

Ibid.

Eddington A. to Shapley H. , 30 December 1918 (Harvard).

Eddington , op. cit. (ref. 52), 168.

Shapley H. and Shapley M. B. , “Studies based on the colors and magnitudes in stellar clusters. Fourteenth paper: Further remarks on the structure of the galactic system”, Astrophysical journal, 1 (1919), 107–40 (p. 126).

Eddington A. to Shapley H. , 30 December 1918 (Harvard).

Wirtz , op. cit. (ref. 31).

van Maanen A. , “Investigations on proper motion, eighth paper: Internal motion in the spiral nebula M94 = NGC 4736”, Astrophysical journal, lvi (1922), 208–16 (p. 215).

Smart W. M. , “The motions of spiral nebulae”, Monthly notices of the Royal Astronomical Society, lxxxiv (1924), 333–53 (p. 352).

Smith , op. cit. (ref. 26), chap. 6. See also Hoskin M. A. and Berendzen R. , “Hubble's announcement of Cepheids in spiral nebulae”, Astronomical Society of the Pacific leaflet, no. 504 (June 1971).

The absolute magnitudes of the novae that had been detected in a few spirals were hotly disputed and so the use of the novae as distance indicators was contentious.

Between 1915 and 1920 Harlow Shapley made an extensive study of the globular clusters. In the course of this study he developed a number of distance indicators which he claimed enabled him to secure distance estimates to the globular clusters, but it was still being debated in 1924 just how accurate Shapley's distances were ( Smith , op. cit. (ref. 26), chaps 5 and 7).

Silberstein L. , “The curvature of de Sitter's space-time derived from globular clusters”, Monthly notices of the Royal Astronomical Society, lxxxiv (1924), 363–6.

Russell H. N. to Shapley H. , 17 September 1920 (Harvard).

Shapley H. to Russell H. N. , 30 September 1920 (Harvard).

Russell H. N. to Shapley H. , 12 October 1920 (Harvard).

Silberstein L. to Shapley H. , 5 January 1924 (Harvard).

Silberstein L. to Shapley H. , 17 April 1924 (Harvard).

Shapley H. to Eddington A. S. , 31 May 1924 (Harvard).

North points out that Silberstein's derivation was analytically correct ( North , op. cit. (ref. 26), 101–4).

Lundmark K. , “The determination of the curvature of space-time in de Sitter's world”, Monthly notices of the Royal Astronomical Society, lxxxiv (1924), 747–70 (p. 767).

Ibid.

Strömberg G. , “Analysis of radial velocities of globular clusters and non-galactic nebulae”, Astrophysical journal, lxi (1925), 353–62 (p. 362).

Lundmark K. , “The motions and distances of spiral nebulae”, Monthly notices of the Royal Astronomical Society, lxxxv (1925), 865–94.

Strömberg , op. cit. (ref. 75), 358.

Dose A. , “Zur Statistik der nichtgalaktischen Nebel auf Grund der Köngistuhl-Nebellisten: Mit einer Bewerkung über der Radialbewegungen der Spiralnebel”, Astronomische Nachrichten, ccxxix (1927), 157–76.

Weyl's hypothesis actually consisted of assuming that the stars (galaxies) can be treated as a uniformly-distributed set of free particles which except for small peculiar motions remain at rest with respect to the spatial coordinates being used, but the proper distances measured by rigid measuring rods will change with time due to a dependence of the metric tensor on time. Thus, a spectral shift is to be expected as light travels from one particle to another ( Tolman , op. cit. (ref. 43), 356–8).

Weyl H. , “Zur allgemeinen Relativitätstheorie”, Physikalische Zeitschrift, xxiv (1923), 230–2 (p. 230).

Lemaître G. , “Un univers homogène de masse constante et de rayon croissant, rendant compte de la vitesse radiale des nébuleuses extra-galactiques”, Annales de la Société Scientifique de Bruxelles, xlvii (A) (1927), 49–56. This was translated as “A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extra-galactic nebulae”, Monthly notices of the Royal Astronomical Society, xci (1931), 483–90.

Robertson H. P. , “On relativistic cosmology”, Philosophical magazine, 7th series, v (1928), 835–48 (p. 844).

Ibid., 845.

Ibid., 836. North writes that: “Robertson is fairly typical of his contemporaries. Whilst he was aware of the logical difficulties in interpreting ‘time’ and ‘distance’ variables, he offered no criterion by which his own particular choice could be justified” ( North , op. cit. (ref. 26), 118). We are not concerned here with the correctness of the mathematical derivations or physical interpretations; our interest is the perception of these analyses by contemporary astronomers.

Hubble E. , “Extra-galactic nebulae”, Astrophysical journal, lxiv (1926), 321–69 (p. 368). Hubble used the formulae for the radius, mass, and volume of the Einstein universe that were given in A. Haas's Introduction to theoretical physics (New York, 1925), ii, 372–4.

Hubble E. , “A relation between distance and radial velocity among extra-galactic nebulae”, Proceedings of the National Academy of Sciences, xv (1929), 168–73. Hubble was well aware of the importance of his paper. In May 1929 he told Shapley that with this investigation a new phase of astronomy was opening ( Hubble E. to Shapley H. , 15 May 1929 (Harvard)).

This was discussed by Hubble , op. cit. (ref. 26), 102–3.

On Humason see Bowen I. S. , “Milton Lasell Humason”, Quarterly journal of the Royal Astronomical Society, xiv (1972), 235–6.

Humason M. , “The large radial velocity of NGC 7619”, Proceedings of the National Academy of Sciences, xv (1929), 167–8 (p. 167).

Hubble , op. cit. (ref. 86), 174.

Ibid., 173.

Hubble E. , “The exploration of space”, Harpers magazine, lviii (1928–29), 732–8 (p. 738).

It is relevant to enquire here if Hubble was aware of the explicit predictions by Weyl, Lemaître and Robertson of a linear redshift-distance relation when he published his 1929 paper. We know that in his research programme Hubble had set himself the task of testing de Sitter's model. In 1929 he had also noted that attempts had been made to explain the presence of a K term as a “correlation between apparent radial velocities and distances, but so far the results have not been convincing” ( Hubble , op. cit. (86), 168). Hubble would have known of the papers of Silberstein, Strömberg and Lundmark and their attempts to secure observationally a redshift-distance relation. It is unthinkable that he had not read them: Strömberg was, after all, at the same observatory as Hubble; and Lundmark's papers on a redshift-distance relation had both appeared in the influential and widely-read Monthly notices of the Royal Astronomical Society, as had Silberstein's “The curvature of de Sitter's space-time derived from globular clusters” (ref. 65). Yet it is not clear from his public pronouncements whether or not Hubble was acquainted with the theoretical studies of Weyl, Robertson and Lemaître . In his 1929 paper Hubble did not mention any mathematical researchers with whom he had been in contact. Also, a letter in the Hubble papers from Robertson H. P. to Hubble , probably dated sometime in 1933, suggests that in 1929 Hubble was not aware of the mathematical predictions of a linear relation, since Robertson listed the predictions for him. Against the evidence of Hubble's paper and Robertson's letter is Hubble's contact with R. C. Tolman. Tolman was a Professor of Physical Chemistry and Mathematical Physics at the California Institute of Technology and he was very well versed in relativity theory. In 1934 his classic text-book Relativity, thermodynamics and cosmology was published and during the 1930s he collaborated with Hubble in studies of galaxies. In 1929 Tolman wrote a paper “On the astronomical implications of the de Sitter line element for the universe” (Astrophysical journal, lxix (1929), 245–74). This paper is dated 25 February. Hubble's paper had been sent to the National Academy of Sciences on 17 January but did not appear in the Society's Proceedings until the 15 March issue. Yet Tolman had emphasised in his paper that one of the three well-established facts about the galaxies was that they exhibited a relation between velocity and distance of the form dλ/λ = α (r/R), where a is a constant. Tolman termed this Hubble's relationship. Clearly he must have known of Hubble's research before Hubble's paper (which we will recall was held for over a year before publication) was published in the Proceedings. How much Tolman knew is unclear: Maybe Tolman and Hubble had discussed Hubble's research programme, or maybe Hubble had informed Tolman of the conclusions of his paper. Perhaps Tolman even told Hubble about the predictions of a linear redshift-distance relation. Such a conjecture is strengthened when we note that Hetherington has pointed out that Tolman was based at Pasadena and that the offices of the Mount Wilson Observatory were cited in Pasadena. Also, in 1928 both Hubble and Tolman had been appointed to an advisory committee that was involved in the planning of the 200–inch telescope that was to be financed by the Rockefeller Foundation. Thus, Hubble had contact with at least one theoretician interested in and aware of the latest developments in relativistic cosmology. Under these circumstances it seems possible that Hubble knew of the theoretical predictions of a linear relation prior to the publication of his 1929 paper. P. J. E. Peebles has surmised that because Lemaître's 1927 value for K (600 km/sec per Megaparsec) was so close to Hubble's 1929 value (500 km/sec per Megaparsec), that “there must have been communication of some sort between the two” (P. J. E. Peebles Physical cosmology (Princeton, 1971), 8). There is, however, no documentary evidence for this; certainly there are no letters in the Hubble papers which corroborate Peebles's guess. Also, if Peebles is correct, then the neglect of Lemaître's paper until 1930 is surprising.

Hetherington N. S. , The development and early application of the velocity-distance relation (unpublished Ph.D thesis, Indiana University, 1970), 130–40.

Ibid., 139.

Hubble E. to Shapley H. , 15 May 1929 (Harvard).

Hubble , op. cit. (ref. 26), 4.

Hubble E. to de Sitter W. , 21 August 1930 (Hubble papers, Huntington Library).

Shapley H. , “Note on the velocities and magnitudes of external galaxies”, Proceedings of the National Academy of Sciences, xv (1929), 565–70.

Ibid., 566.

Ibid., 567.

Hubble had mistaken what are now identified as H II regions—bright clouds of ionized gas—for the brightest stars. Since these clouds are intrinsically more luminous than stars, it meant that the corresponding distances were underestimated. This was, however, only demonstrated convincingly much later and within the astronomy of the period Hubble's assertion was entirely reasonable. See Sandage A. , “Current problems in the extragalactic distance scale”, Astrophysical journal, cxxvii (1958), 513–26 (p. 522).

Shapley H. to Hubble E. , 7 May 1929 (Harvard).

Hubble E. to Shapley H. , 15 May 1929 (Harvard).

Ibid.

Ibid.

Shapley H. to Russell H. N. , 22 May 1929 (Harvard).

Shapley H. , “Progress in extragalactic explorations”, notes on a symposium address to the American Philosophical Society, 26 April 1930 (Harvard).

de Sitter W. , “On the magnitudes, diameters and distances of the extragalactic nebulae, and their apparent radial velocities”, Bulletin of the Astronomical Institute of the Netherlands, v (1930), 157–71 (p. 169). Eddington had told de Sitter of Lemaître's 1927 paper a few weeks before he completed this paper.

Nevertheless, Hubble wanted to ensure that de Sitter recognised that the formulation, testing, and confirmation of the velocity-distance relation, was a “Mount Wilson contribution” since he insisted to de Sitter that “our preliminary note in 1929 was the first presentation of the data where the scatter due to uncertainties in distances was small enough as compared to the range in distances, to establish the relation” (Hubble to de Sitter, 21 August 1930 (Hubble papers, Huntington Library)). Hubble was also surprised that de Sitter had assigned equal weights to his own and Lundmark's distance estimates. In addition, as part of the programme to test the velocity-distance relation, Humason had measured the radial velocities of a number of galaxies and the values which Humason had secured were then published in the Mount Wilson Annual Reports. De Sitter had used these radial velocities in his own derivation of the linear relation. “We have always assumed”, Hubble protested, “that, where a preliminary note is published and a program is announced for testing the result in new regions, the first discussion of the new data is reserved as a matter of courtesy to those who do the actual work. Are we to infer that you do not subscribe to this ethics; that we must hoard our observations in secret? Surely there is a misunderstanding somewhere?” (Hubble to de Sitter , ibid.). Their later correspondence was amicable, but de Sitter, as he revealed to Shapley, was upset by this letter ( Shapley H. to Russell H. N. , 26 November 1931 (Harvard)).

Tolman R. C. , “On the astronomical implications of the de Sitter line element for the universe”, Astrophysical journal, lxix (1929), 245–74 (p. 245).

Report of the meeting of the Royal Astronomical Society of 10 January 1930, The observatory, liii (1930), 33–44 (p. 38).

Friedmann A. , “Über die Krümmung des Raumes”, Zeitschrift für Physik, x (1922), 377–86.

Tolman R. C. , “On the possible line elements for the universe”, Proceedings of the National Academy of Sciences, xv (1929), 297–304. In this paper, communicated on 13 March 1929, Tolman said that “it should be noted that our assumption of a static line element takes no explicit recognition of any universal evolutionary process which may be going on. The investigation of non-static line elements would be very interesting” (p. 304).

Op. cit. (ref. 112), 39.

McVittie G. C. , who in 1929 was a research student of Eddington's, recalls that “I well remember the day when Eddington, rather shamefacedly, showed me a letter from Lemaître which reminded Eddington of the solution to the problem which Lemaître had already given. Eddington confessed that, though he had seen Lemaître's paper in 1927, he had completely forgotten about it until that moment” ( McVittie G. C. , “Georges Lemaître”, Quarterly journal of the Royal Astronomical Society, viii (1967), 294–7 (p. 295)). Early in 1930 Eddington sent de Sitter a copy of Lemaître's 1927 paper. Across the top of the front page Eddington wrote: “This seems a complete answer to the problem we were discussing” (Leiden Observatory Archives). There seems little doubt that the “problem” was to find a suitable alternative to Solutions A and B.

de Sitter W. , “On the distances and radial velocities of extra-galactic nebulae, and the explanation of the latter by the relativity theory of inertia”, Proceedings of the National Academy of Sciences, xvi (1930), 474–88 (p. 482).

de Sitter W. to Shapley H. , 17 April 1930 (Harvard).

Report of the British Association for the Advancement of Science (1931), 584.

New York Times, 3 January 1931, 1.

Einstein A. and de Sitter W. , “On the relation between the expansion and the mean density of the universe”, Proceedings of the National Academy of Sciences, xviii (1932), 213–14.

North , op. cit. (ref. 26), 111. See also pp. 111–21.

Tolman , op. cit. (ref. 43), 347. Hetherington has argued that circumstances of publication may also have contributed to the neglect of the models of the expanding universe ( Hetherington N. S. , “The delayed response to suggestions of an expanding universe”, Journal of the British Astronomical Association, lxxxiv (1973), 22–28).

Hubble E. and Humason M. , “The velocity-distance relation among extra-galactic nebulae”, Astrophysical journal, lxxiv (1931), 43–80. This paper is described in more detail than here by Hetherington , op. cit. (ref. 94), 144–52.

Hubble and Humason , op. cit. (ref. 124), 77.

Humason M. , “Apparent velocity-shifts in the spectra of faint nebulae”, Astrophysical journal, lxxiv (1931), 35–42 (p. 37).

Shapley H. , Star clusters (New York, 1930), 189.

By 1935 even the careful Hubble felt able to write that “the velocity-distance relation is so firmly established that it is assumed to be true for all nebulae, and the observed residuals are analysed for the information they give concerning the scatter in the intrinsic luminosities or the luminosity function of nebulae” ( Hubble , op. cit. (ref. 26), 120).

Jeans J. H. , Astronomy and cosmogony (Cambridge, 1928), 409.

Jeans , op. cit. (ref. 35), 81.

The “Hubble constant” is K in our notation.

Eddington A. S. , The expanding universe (paperback edition, Ann Arbor, 1952), 68.

Ibid., 8.

North , op. cit. (ref. 26), 228.

Hubble , op. cit. (ref. 86), 173.

Hubble E. to Shapley H. , 15 May 1929 (Harvard).

Humason , op. cit. (ref. 126).

Hubble E. to de Sitter W. , 23 September 1931 (Huntington Library). Of course by 1931 de Sitter's model B had dropped completely from favour and Hubble could not claim that the linear redshift-distance relation demonstrated the reality of the de Sitter effect.

Shapley , op. cit. (ref. 108). The seeds of doubt that were to blossom into this declaration had been sown at least five years earlier when he had asked Doig: “What do these line shifts mean, anyway? Personally, I think they mean velocities, and yet…” ( Shapley H. to Doig P. , 1 December 1926 (Harvard)).

North , op. cit. (ref. 26), 229–34, and Hetherington , op. cit. (ref. 94), 166–71.

de Sitter W. , “On the expanding universe”, Proceedings of the Royal Academy of Sciences of Amsterdam, xxxv (1932), 596–607.

North , op. cit. (ref. 26), 224.

Jeans , op. cit. (ref. 35), 327.