Newton,Providence and the Universe of Stars

Voltaire , Elemens de la philosophie de Neuton (Amsterdam, 1738), 383.

“Continuo opus esse miraculo ne Sol et fixae per gravitatem coeant”, David Gregory memorandum no. 44 (Royal Society Library), available in The correspondence of Isaac Newton [hereafter: Newton, Correspondence], ed. by Turnbull H. W. (Cambridge, 1959–), iii, 334.

The first evidence of ‘proper motions’ of stars was given by Halley, “Considerations on the Change of the Latitudes of some of the Principal Fixt Stars”, Philosophical transactions, xxx (1717–19), 736–8.

On the subject of Newton and providence, see Cohen Bernard I. , “Isaac Newton's Principia, the Scriptures, and the Divine Providence”, in Philosophy, science, and method, ed. by Morgenbesser Sidney (New York, 1969), 523–48; Guerlac Henry and Jacob M. C. , “Bentley, Newton and Providence”, Journal for the history of ideas, xxx (1969), 307–18; Kubrin David , “Newton and the Cyclical Cosmos: Providence and the Mechanical Philosophy”, ibid., xxviii (1967), 325–46; and Jacob Margaret C. , The Newtonians and the English Revolution (Hassocks, Sussex, 1976).

The history of De systemate is discussed in Cohen I. B. , Introduction to Newton's ‘Principia’ (Cambridge, 1971), 109–15 and 327–35.

In the familiar Motte-Cajori translation of the Principia (Berkeley & Los Angeles, 1934), 596–7.

I discuss this in “The English Background to the Cosmology of Wright and Herschel” in Cosmology-history-theology, ed. Yourgrau Wolfgang (in press). In the same article I describe the published works that form the context to the manuscripts and letters analysed in the present paper.

Principia, ed. Motte-Cajori , 597: “Tantis igitur intervallis ab invicem distantia sidera nec trahent se mutuo sensibiliter, nec a Sole nostro trahentur” (University Library, Cambridge, Add. MS 3990, f35^r).

Newton , Principia (London, 1687), 420; ed. Motte-Cajori , 422.

Copies of the printed versions of the relevant sermons, and of the 1756 edition of the four letters from Newton to Bentley, are conveniently available with an introduction by Perry Miller in Cohen I. B. (ed.), Isaac Newton's papers and letters on natural philosophy [hereafter: Cohen, Newton's papers and letters] (Cambridge, 1958). A more accurate text of the letters from Newton and the one surviving letter from Bentley to Newton are to be found in Newton , Correspondence, iii.

Bentley's seventh sermon (and the second on “A Confutation of Atheism from the Origin and Frame of the World”) (London, 1693), 4; Cohen , Newton's papers and letters , 316. Cf. Bentley to Newton, 18 February 1692/3, Newton , Correspondence, iii, 246.

Most notably in the General Scholium in the second (1713) and later editions of the Principia.

Cohen , Newton's papers and letters, 281–2; Newton , Correspondence, iii, 234.

Cohen , Newton's papers and letters, 292–3; Newton , Correspondence, iii, 238.

Cohen , Newton's papers and letters, 293; Newton , Correspondence, iii, 239. In the printed version of his seventh sermon, Bentley criticizes the position he originally held: Cohen , op. cit., 351–2.

Cohen , Newton's papers and letters, 295–6; Newton , Correspondence, iii, 239.

See for example Charlier C. V. L. , “How an Infinite World may be Built Up”, Arkiv für Matematik, Astronomi och Fysik, xvi, no. 22 (1922).

Seeliger H. , “Ueber das Newton'sche Gravitationsgesetz”, Astronomische Nachrichten, cxxxvii, no. 3273 (1 February 1895), cols 129–36.

Newton , Correspondence, iii, 246–53.

Ibid., 250–1. The second paragraph, being a comment, is in square brackets in the original.

Cohen , Newton's papers and letters, 351.

The drafts are in the first ‘Principia box’, University Library, Cambridge, Add. MS 3965; all manuscript references are to this box unless otherwise indicated.

f280^v: “Stellae fixae quiescunt et immensis tam ab invicem quam a Sole nostro intervallis distant.”

ff280^v–280^r: “Datas enim servant positiones ad Aphelia & Nodos ideoque quiescunt: Et nisi quiescerent non manerent in coelis suis. Nam cum nulla sit earum parallaxis sensibilis ex Terrae motu annuo oriunda, necesse est ut ingens sit earum distantia a systemate nostro, et nulla est vis naturae qua corpora tam longinqua in orbibus circa Terram revolventia retineri possi[n]t & a motibus rectilineis retrahi. Abirent igitur in tangentibus orbium.” The last two words may be seen in Figure 1. Note that in these transcriptions of Newton's Latin, occasional phrases are lacking because of damage to the original MS but these can usually be supplied with complete certainty from other drafts. A manuscript intimately related to the sequence we are now considering, namely f175^r, consists of rough drafts of proofs that the stars are very distant, and is notable for its discussion of the ‘double star’ method of detecting annual parallax that was to be so important in the nineteenth century: “Ingens fixarum distantia concluditur ex defectu parallaxeos orbis magni. Nam Fixae majores atque adeo propinquiores respectu minimarum illarum quae solis Telescopijs videntur nullum habent ex parallaxi motum sensibilem.”

Which minimum distance he argues (f279^v) is 12,000 times the distance of the Earth from the Sun.

Gregory J. , Geometriae pars universalis (Padua, 1668), 148. Cf. de Villamil R. , Newton: The man (London, [1931]) 106.

f279^v: “Unde facile colligitur quod Sol distantia ejus a Terra 900000 vel numero rotundo 1000000 vicibus circiter augeretur.”

ff279^v–279^r.

f279^r: “… sed ejusmodi minutias negligo.”

For the original draft, see Figure 1.

Kepler's discussion (Epitome, Book I, Part II) is quoted by Koyré Alexandre in From the closed world to the infinite universe (Baltimore, 1957), chap. 3. Twelve is the maximum number of stars or points satisfying our conditions. Cf. Newton , Correspondence, iii, 321.

f275^r.

f276^r.

f74^r (see Figure 2), continued on the verso.

In his copy of Wing's Vincent Astronomia Britannica (London, 1669), now in the Library of Trinity College, Cambridge, Newton has written at the end of the catalogue of fixed stars (p. 263 of the tables): Magnitudinis primae 16 secundae 55 iuxta Hal. vel 62 juxta Wing tertiae 227 quartae 506. Several of the figures have been corrected, suggesting that Newton counted and rechecked the addition himself, and these may be the figures Newton now uses. Turnbull H. W. (Newton, Correspondence, ii, 394) draws attention to ULC Add. MS 3958, ff38^r–40^v, where Newton has compiled “A Table of ye fixed Starrs for ye yeare 1671. Of ye three first Magnitudes”, with totals of 13, 43 and 174 respectively. We shall encounter other rough totals in our discussion of “Related Newton documents”, below.

The parallel between the path trodden in secret by Newton and that followed publicly by Herschel a century later is astonishing. Herschel in 1781 adopted two postulates: “1. Let the stars be supposed, one with another, to be about the size of the Sun. 2. Let the difference of their apparent magnitudes be owing to their different distances, so that a star of the second, third, or fourth magnitude is two, three, or four times as far off as one of the first” (“On the Parallax of the Fixed Stars”, Philosophical transactions, lxxii (1782), 82–111, pp. 104–5). When challenged on the second postulate, Herschel added a remark that he “rather meant the order into which the stars ought to be distinguished than that into which they are commonly divided” (ibid., 105); Newton claims his predictions would match the observations “si stellae in magnitudines juxta proportione lucis inversa quadratorum numerorum progressione 1, 1/4, 1/9, 1/16, 1/25, 1/36, 1/49 designatas distinguerentur” (f74^r, our Figure 2). Near the end of his life, when he had hit upon the technique of using two exactly similar telescopes fitted with a range of diaphragms in order to measure the relative brightness of stars, Herschel could at last afford to put at risk the uniform distribution model plus second postulate, by testing it against the numbers of stars of successive magnitudes (“Astronomical Observations, and Experiments Tending to Investigate the Local Arrangement of the Celestial Bodies in Space”, Philosophical transactions, cvii (1817), 302–31). In the course of an elaborate argument, he considers spheres surrounding the Sun at distances 1, 3, 5, … and assigns the space between the first two spheres to first-magnitude stars, between the next two to second-magnitude stars, and so on. He then expects numbers of stars of successive magnitudes to be equal to the differences between successive pairs in the sequence 1, 27, 125, …; that is, to 26, 98, 218, …, which is a sequence Newton tries on f280^r (Figure 1)! Cf. my William Herschel and the construction of the heavens (London, 1963), chaps 2 and 5.

Halley Edmond , “The Number, Order, and Light of the Fix'd Stars”, Philosophical transactions, xxxi (1720–21), 24–26; the relevant Journal Book of the Royal Society shows that the paper was read on 16 March 1720/1; that is, 1721, and not 1720 as usually stated. On the derivation of the modern definition, see Pannekoek A. , A history of astronomy (London, 1961), chap. 40.

The counting of numbers of stars of successive magnitudes is important in modern astronomy, for if the counts are lower than expected this may indicate the presence of a dark absorbing cloud in interstellar space. Modern astronomy defines magnitudes so that a first-magnitude star shall continue to be 100 times brighter than a sixth-magnitude star, in conformity with the approximate values derived from the traditional classification. Intermediate magnitudes are then defined so that the ratios of the corresponding figures for successive magnitudes are always the same: That is, ⁵√100 or 2·512. The distances represented by the magnitudes 1, 2, 3, 4, 5, 6, found by taking the square root of the brightnesses, are then approximately 1, 1·59, 2·51, 3·98, 6·31, 10·00, so that the distances are sometimes less and sometimes greater than the corresponding magnitudes, and for the modern magnitude 3 the distance is substantially less than 3. However, the mean modern magnitudes for stars listed by Ptolemy as of magnitude 2, 3, 4 are 2·21, 3·28 and 4·35 respectively ( Lundmark Knut , “The Estimates of Stellar Magnitudes by Ptolemaios, Al Sûfi and Tycho Brahe”, Vierteljahrsschrift der Astronomischen Gesellschaft, lxi (1926), 230–6, p. 232), so that, caeteris paribus, the corresponding distances will be greater and the numbers of stars larger than would otherwise be the case. Of course, stars actually vary enormously in luminosity, and a seemingly-bright star may be many times more distant than a faint one, so that it is in fact hazardous to draw conclusions along these lines until the numbers of stars involved are much larger.

Halley Edmond , “Of the Infinity of the Sphere of Fix'd Stars”, Philosophical transactions, xxxi (1720–21), 22–24. This was read on 9 March 1720/1, “the President in the Chair”. The summary in the Journal Book shows that considerations of the physical nature of light played a greater role in Halley's thinking than the published paper suggests: “The other objection against an infinite number of stars is from the small quantity of light which they all give whereas were there an infinite number it should seem to be much more. To this Dr Halley replies that light is not divisible in infinitum and consequently when the stars are at very remote distances their light diminishes in a greater proportion than according to the common rule and at last becomes intirely insensible even to the largest telescopes” (Journal Book XII (1720–26) (copy), 94). A full account of the so-called ‘paradox’ is given in Jaki S. L. , The paradox of Olbers' Paradox (New York, 1969), and a shorter discussion in Hoskin M. A. , “Dark Skies and Fixed Stars”, Journal of the British Astronomical Association, lxxxiii (1973), 254–61.

For this and the preceding quotations, see Figure 2.

f275^v.

ff184^r, 184^v.

A fourth manuscript (ULC Add. MS 4005, ff23^r et seq.) is headed “The Mechanical Frame of the World” and the second paragraph reads: “2. The Sun is a fixt star & the fixt stars are scattered throughout all the heavens at very great distances from one another & rest in their several regions being great round bodies vehemently hot & lucid & by reason of the great quantity of their matter they are endued with a very strong gravitating power.”

f542^v.

Published Cambridge, 1962. In a modern printed edition the texts have of course a polished appearance alien to the originals.

ULC Add. MS 4005, ff21^r–22^r; Hall & Hall, Unpublished papers of Newton, 374–6.

ULC Add. MS 4005, ff45^r–49^r; Hall & Hall, Unpublished papers of Newton, 378–85.

Pannekoek , op. cit. (ref. 37), 446. For reasons explained above (ref. 38), the modern definition defines the ratio between successive magnitudes as 2·512. “Suppose we count all stars brighter than successive magnitudes, say 10, 11, 12, etc., in a certain area of sky. If the stars were uniformly distributed in space, the numbers should increase for each magnitude step by a factor of about 4·0. The inverse-square law of brightness requires that two groups of stars differing by one magnitude should have average distances proportional to the square root of their relative brightnesses, or √2·512. Then the volume of space probed by the counts should increase in proportion to the cube of the distance, or (√2·512)³ = 3·98, which, for a transparent, uniformly populated space should equal the ratio of the numbers of counted stars per magnitude” ( Menzel D. H. Whipple F. L. and de Vaucouleurs G. , Survey of the universe (Englewood Cliffs, N.J., 1970), 603).

On the ‘miracle’, see ref. 2 above. On James Gregory's method and the model of a uniform distribution of stars, see Gregory David memorandum no. 33 (Royal Society Library, available in Newton, Correspondence, iii, 312), item 13 (which includes a numerical slip); there is no known evidence that David had prior knowledge of James's method. On the relevance of Book I, Prop. LXX, see the Notae by David Gregory on the first edition of the Principia (Royal Society Library), where against Book III, Prop. XIV, Corol. 2, he writes: “Sed et praeter immensam distantiam earum positio circum circa effectus impedit ex prop: LXX lib: 1” (“But besides the huge distance, their location on all sides hampers the effects, by Book I, Prop. LXX”). This closely parallels Newton's own addition to Corollary 2 in the second edition: “Quinimo fixae in omnes coeli partes aequaliter dispersae contrariis attractionibus vires mutuas destruunt, per prop. LXX lib. 1.” I owe to Miss Christina Eagles references to two other occasions on which Gregory touches on the problem of the stars and gravity. The first is in a memorandum which mentions “Mr Newton's exceptions against my book” (the Elementa of 1702), but which is mainly on other topics and concludes: “The fixt Starrs may move inter se by their mutual actions” (f76^r of the David Gregory memoranda in the Royal Society Library). The second is a slip of paper pasted into Gregory's Notae (f47), headed “Ad prop: VII. Lib. III”. After ascribing an understanding of gravity to ancient writers (cf. ref. 51), the text continues: “Verum si corpora omnia in omnia sint gravia, Quidni Stellae fixae ex gravitate coeant et concurrant? An continuo opus est miraculo ad hunc effectum impediendum? an in immensa quae intercedit inter eas distantia, languescit gravitas? an potius circa diversa centra rotatae planetarum more revolvuntur.” Gregory has subsequently added: “Si mundus esset finitus obtineret haec obiectio: Existente vero infinito vim nullam obtinet.”

Gregory David , Astronomiae physicae et geometricae elementa (Oxford, 1702), 159–60, transl. from the second English edn (London, 1726), 288–90.

Gregory , Elementa , 483; transl. from second English edn, 856. As is well known, Newton allowed Gregory to include in his Preface material drafted by Newton in the 1690s in which Newton credits the ancients with profound scientific and philosophical insight. In one surviving manuscript (f278^r) Newton understands Lucretius, De rerum natura, 1, 984–91, in the sense of this Gregory passage, in a sentence he afterwards cancels: “… in materiam omnem circum circa positam et per gravitatem mundi infiniti in aequilibrio sustinentur ne se mutuo ruant.”

Halley , op. cit. (ref. 39) 23.

Halley , op. cit. (ref. 3).

However, examination of the manuscript (Royal Society Library) shows that the word “nearly” before “in aequilibrio” is an interpolation into the original draft. This, placed alongside the summary in Journal Book XII (copy), 93, which has him say that in an infinite system a star is at rest “because it has not any tendency to move one way rather than another”, suggests that Halley may simply have thought that stars in an infinite system initially at rest will automatically remain at rest.

Newton , Optice (London, 1706), Qu. 20: “… Et Quidnam est quod impedit, quominus Sol & Stellae fixae in se mutuo irruant?”.

For Latin text, see ref. 49. A draft on f236^v gives a slightly fuller version: “Quinetiam fixae in omnes coeli partes aequaliter dispersae contrarijs attractionibus vires mutuas destruunt. Nam corpus intra superficiem sphaericam constitutum nullam in partem a viribus superficiei attrahitur per Prop LXX Lib 1.”

f362^r: “… tantae autem sint distantiae fixarum et a Sole et a seinvicem ne systemata eorum in se mutuo cadant” (the draft is transcribed in Hall & Hall, Unpublished papers of Newton, 355–9); f152^v: “… removendo stellas fixas ad commodas distantias ne cadant in seinvicem” (cf. Cohen , op. cit. (ref. 4), 531, where however the draft is said to be for Prop. XLI; it is for the very end of Prop. XLII which leads into the General Scholium).

“Elegantissima Haecce solis, planetarum & cometarum compages non nisi consilio & dominio entis intelligentis & potentis oriri potuit…. Et ne fixarum systemata per gravitatem suam in se mutuo cadant, hic eadem [ad] immensam ab invicem distantiam posuerit” (p. 527).

Newton , Optice (London, 1706), Qu. 23: “Nam cum Cometae moventur in Orbibus valde eccentricis, undique & quoquoversum in omnes coeli partes; utique nullo modo fieri potuit, ut caeco fato tribuendum sit, quod Planetae in orbibus concentricis Motu consimili ferantur eodem omnes; exceptis nimirum irregularitatibus quibusdam vix notatu dignis, quae ex mutuis Cometarum & Planetarum in se invicem actionibus oriri potuerint, quaeque verisimile est fore ut longinquitate temporis majores usque evadunt, donec haec Naturae Compages manum emendatricem tandem sit desideratura.” The Latin is more tentative than the English.

Newton , Correspondence, iii, 235; Cohen , Newton's papers and letters, 286–7.

Alexander H. G. (ed.), The Leibniz-Clarke correspondence (Manchester, 1956), 14.