होम Philosophical Review Criticism in the History of Science: Newton on Absolute Space, Time, and Motion, I

Criticism in the History of Science: Newton on Absolute Space, Time, and Motion, I

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खंड:
68
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english
पत्रिका:
The Philosophical Review
DOI:
10.2307/2182544
Date:
April, 1959
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Philosophical Review

Criticism in the History of Science: Newton on Absolute Space, Time, and Motion, I
Author(s): Stephen Toulmin
Source: The Philosophical Review, Vol. 68, No. 1 (Jan., 1959), pp. 1-29
Published by: Duke University Press on behalf of Philosophical Review
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CRITICISM IN THE HISTORY OF SCIENCE:
NEWTON ON ABSOLUTE SPACE, TIME,
AND MOTION, I
N looking back at the scientific ideas of the past how critical
can we afford to be? Are we entitled to sit in judgment on
our great scientific predecessors and to award them good or
not-so-good marks for their achievements; or should we rather
adopt an aesthetic approach, limit ourselves to the sensitive
reconstruction and appreciation of their ideas, and, having
pieced together (like some antique vase) the thoughts of a Kepler,
a Boyle or a Cuvier, pass reverently on to the next Old Master?
There is here, at any rate on the surface, a conflict of interests
between the historian and the philosopher. To the historically
minded the second, aesthetic approach has great attractions.
For those who presume to judge, criticize, assess, run great
risks of anachronism. It is so easy, for example, to wri; te off
Aristotle's Physics as a failed attempt at producing Newton's
Principial and to allow the changes in style which have affected
I

1 The major part of Aristotle's discussion of locomotion in the Physics
consists of conceptual analysis rather than of physical representation and
explanation: he is concerned to show that his notions are logically coherent,
not that they "save the phenomena"-hence,
among other things, his long
discussion of Zeno's paradoxes. One short chapter (VII, v; 249b 27-25ob 7)
contains something which at first sight resembles modern kinematics, and on
this passage is based the idea that for Aristotle force was proportional to velocity. But two things should be noted: (a) so far from formulating any mathematical function or equation relating force and velocity, Aristotle does not even
employ any word for "speed" or "velocity," and his word "&vva/uss" can only
very dubiously be rendered as "force"; and (b) since he is talking about
steady-state motion against resistances, the law which his discussion foreshadows is not Galileo's principle of inertia but Stokes's law, viz., that the terminal
velocity of a body moving through a resisting medium is proportional to the
force propelling the body and inversely proportional to the viscosity of the
medium.
From his examples, which have to do with gangs of men shifting
boats and the like, it seems that he would best be translated (e.g.): "If a
given effort is needed to shift a load so far in such a time (and half as far in half
the time), then half the strength will shift half the load the same distance in the
same time" (25oa 5-7). Neither Sir Thomas Heath (Mathematics in Aristotle,
nor I. E. Drabkin (A Source-Bookof Greek Science,
Oxford, I949, pp. I43-I46)
I
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STEPHEN TO ULMIN

physical thought in the last century to blind one to the solid
merits of the caloric theory;2 or alternatively to find in some
early, passing speculation about, say, "atoms" merits, insight,
even prevision, of a kind which are simply not there.3 Critical
anachronism of this kind is, it seems, as much of a danger as that
grosser sort which finds in Anaximander, for example, the doctrine
that "living things had arisen from inorganic matter."4 And
some would argue that we can hope to avoid this danger only
if we remain scrupulously neutral and are content to discover and
depict the ideas of earlier scientists as faithfully as we can,
without attempting to award marks, to pass judgment, or to
give (at any rate in our own terms) a critical assessment of their
achievements.
The philosopher of science, on the other hand, finds it hard
to renounce so completely his critical ambitions. For his aim is
to understand the nature of the scientist's inquiries and to refine
his critical categories by comparison with the great exemplars
of scientific theory. He must not, it is true, let his philosophical
prejudices do violence to historical justice by demanding of all
New York, 1948, pp. 203-204,
and AmericanJournal of Philology,LIX [I938],
60-84) avoids entirely the anachronism of looking for functional relations of a
modern kind in Aristotle's Physics. (I have had valuable discussions with
Mr. G. E. L. Owen about this point.)
2 The change-over to the kinetic theory of matter did result in one striking
theoretical economy, since thermal phenomena could now be explained in
terms of the properties, not of a duplicate matter existing alongside (or within)
the ponderable matter of gravitational theory, but of the very same matter
and its motions. But it did not at once lead to the explanation of many
new phenomena for which the caloric theory was unable to account. Certainly
Rumford's cannon-boring experiment was not crucial against the caloric
theory. Much of the heat-theory of the early nineteenth century could in
fact be rewritten simply in the new idiom, "caloric-flow" being replaced by
"energy-flow," notably Carnot's proof of the Second Law of Thermodynamics.
3 For the Greek philosophers, atomic theories were attractive for logical
or epistemological reasons as often as for scientific ones: the modern
counterpart would be Russell's "logical atomism," for which see "The Philosophy of Logical Atomism" (i9i8),
reprinted in Logicand Knowledge(London,
I956) and discussed by J. 0. Urmson in Philosophical
Analysis (Oxford, I956).
F. M. Cornford was particularly severe on the idea that Epicurus was a
scientist in the modern sense of the term: cf. PrincipiumSapientiae(Cambridge,
I952),
esp. pp. 26-28.
4 Thus S. F. Mason, A Historyof the Sciences(London, 1953), p. I5.
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NEWTON ON ABSOLUTE SPACE

theories, say, that axiomatic structure which few but Euclid's
and Newton's ever quite achieved.5 Still, his very attempt to
interpret and analyze the scientist's role in our intellectual
economy will carry with it criteria of success or failure and will
lead him to look in every theory or notion, present or past, for the
merits characteristic (on his view) of a fruitful scientific idea.
If the merits he chooses to look for are not the appropriate ones,
so much the worse for him; but if his conception of science is
anything like adequate, it will find a place of honor for the great
theories of the past, and will do something to indicate what in
them deserved to last and why the remainder did not succeed
in the struggle for survival between theories.6
The philosopher may even wonder whether the historian's
ideal of neutrality can in fact be achieved, to say nothing of
being desirable. Can one give an account of any scientificdevelopment without being committed to some standards of what is
good or bad in science? Is not the historian of scientific ideas
bound to have his views about the aims and functions of scientific
inquiry and to select and order his quotations-to say no morein accordance with these ideas? The historico-aestheticapproach
has, in any event, dangers of its own as grave as those of anachronism. For suppose one does make the attempt at neutrality,
and resolutely refrains from asking what in the writings of a
scientist of the past was good science, what was bad science, and
what was not science at all. One is then in danger of overlooking
in the scientist one is studying logical or philosophical distinctions
vital to an understandingof his work. Let our historical inquiriesbe just, let them avoid reading into earlier theories scientific
distinctions which grew up only later; but that does not mean
5Whole-hearted acceptance of the "hypothetico-deductive" account of
scientific theories carries with it always this temptation. Cf. R. B. Braithwaite,
ScientificExplanation(Cambridge, I953);
and the various works of J. H.
Woodger, which illustrate both the virtues and the limitations of the "axiomatic
method."
6As contrasted with (e.g.) Rudolf Carnap, who regards the fact that one
cannot apply his "inductive logic" to sophisticated theories in physics as a
sign that these theories have not been completely and exactly formulated"according to the rigorous standards of modern logic." See LogicalFoundations
of Probability(Chicago, I950), pp. 242 if.

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STEPHEN TO ULMIN

they should be critically naive, or Laodicean, or unphilosophical.
By all means let us understand how past ideas looked to those
who held them; but if we are now in a position to see more
clearly, in retrospect, the effective scope and character of a
doctrine or the cross-purposes underlying a dispute, we need not,
surely, in our pursuit of historical purity and fidelity, deprive
ourselves of the benefits of this clearer vision. To import onto
the stage of fourteenth-century dynamics characters from
twentieth-century relativity theory would indeed be an anachronism, but there is no reason why we should not from time
to time view that stage through twentieth-century opera glasses.
I
These reflections on method will be illustrated here by a
re-examination of the notion of absolute space as it appears in
Newton's Principia mathematicsphilosophiae naturalis.7 One needs
almost to apologize for adding to the literature on Newton.
But he is such a key figure in the development of modern science
that his ideas must always be kept in the sharpest focus; his
theories set the pattern for physical science for two centuries,
and provide the philosopher of science with a touchstone for
testing his ideas; and a man who appears at first sight so complex
a mixture of the keen-sighted natural scientist and the naif
philosophical theologian presents historian and philosopher alike
with delicate and fascinating problems. It is therefore no wonder
that he has been the object of so much attention and that his
ideas have attracted commentators of every sort from the axgrinding to the over-sympathetic.
In any case we are only now sufficiently far from his mountainous achievements to see them in proportion-at a Darjeeling to
his Everest, so to speak. During the eighteenth and nineteenth
centuries "Newton" became a sort of lay-figure, personifying
that cool scientific reason which the Enlightenment esteemed and
the Romantics-notably
Blake8-reacted against; Sir Isaac the
7This will be cited in the edition by Florian Cajori (Berkeley, 1934) and
abbreviated as Princ.
8 There is a famous engraving by Blake called "Newton," and several of
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NEWTON ON ABSOLUTE SPACE

man remained only a shadow. Then, after two hundred years,
the temper changed. In physics many of his ideas have been
amended if not entirely superseded; on the logical front his
theories have been subjected by Mach to a rigorous criticism,
some of it (I shall argue) quite anachronistic.9 And those historians who have at last looked behind the traditional figure to
the man found him, not surprisingly, more a creature of his time
and an introverted personality, and less a universal, timetranscending, all-rational genius than he had earlier been painted.
During the last half-century, therefore, historians have tended
increasingly to emphasize the less rational side of his mind.
Maynard Keynes declared that he was not "the first of the age
of reason" but "the last of the magicians" ;10 A. J. Snow has
explored the background of his thought in Jacob Boehme and
other mystical writers;"1 his Rosicrucian connections have been
pointed out;12 his work on chemistry has been so presented as to
imply that he was a Faust rather than a Boyle in his methods,
his interest in alchemical writings being cited as a proof of this ;13
Helene Metzger followed out the influence on eighteenth-century
natural theology in Britain of his idea of universal attraction ;14
his writings refer scathingly to the great mathematician. Goethe's dislike of
Newton's theory of color is also well-known.
9 I use the fifth edition of the translationby T. J. McCormackof Die Mechanik
in ihrer Entwicklung(Leipzig, I 883) published as The Scienceof Mechanics
(Chicago and London, 1942).
'O J. M. Keynes, "Newton the Man" in TheRoyalSociety:NewtonTercentenary
Celebrations(Cambridge, 1947), esp. pp. 28-29.
11 A. J. Snow, Matter and Gravityin Newton's Physical Philosophy(London,
1926).
12
p.

Most recently by E. N. da C. Andrade, Sir Isaac Newton (London,

I954),

I28.

13 Though anybody undertaking a serious experimental study of chemical
phenomena in the middle of the seventeenth century would have been illadvised to ignore the alchemical tradition. The work of Holmyard, Kraus,
Stapleton, and others has made it clear how far the Islamic alchemists were
the forerunners as much of the medieval Western metallurgists as of the
more fanciful "adepts." How far Newton read his alchemists for their practical
experience, how far with cloudier hopes and visions, we shall not know until
his chemical papers have been published and evaluated. For a sample, see
F. Sherwood Taylor's paper in Ambix,V (I956), 59-84.
14 H. Metzger, Attractionuniverselle
et religionnaturellechez quelquescommentateursAnglais de Newton (Paris, 1938).

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STEPHEN TO ULMIN

while E. A. Burtt's famous book on The Metaphysical
Foundations
of
ModernScienceboth helped to establish the general tendency
by its very title, which implied a closer relation between science
and metaphysics than the nineteenth century had acknowledged,
and at -the same time brought to light certain resemblances
between the theology of the Cambridge Platonists, notably
Henry More, and Newton's own general system of ideas.15
Most recently M. Alexandre Koyre has filled in the shading
of the new picture in his book Fromthe ClosedWorldto theInfinite
Universe;16indeed, he has gone rather further than most of his
forerunners thought to do. To Henry More and Joseph Raphson
it seemed that Space must be one aspect of the Deity, namely,
His immensity; both God and Space were (they argued) infinite,
eternal, incorporeal, and immutable, and it was not to be supposed that the Universe could harbor two separate, independent,
and distinct entities having all these characteristics in common.1
More was an older contemporary of Newton's at Cambridge,
Raphson a younger mathematician and disciple. Newton's
own teachings about absolute space and time were already
being labelled "metaphysical" by Mach at the end of the nineteenth century.18 Koyr6 therefore concludes that Newton's Space
and More's are identical, so that in putting forward his notion of
"absolute, true, mathematical duration and space" Newton was
arguing on behalf of the selfsame concepts for which Henry More
fought his long-drawn-out and relentless battle against Descartes."9
This last touch, though new, only rounds off the current picture
of Newton's views, completing what since Mach and Einstein
has become the received doctrine on the subject of Newtonian
space and time.
Yet by this time-and this will be my main thesis-the reaction
has been overdone. The twentieth-century picture of NewtonFaust is becoming by now as exaggerated as the earlier one of
15 E. A. Burtt, The Metaphysical
Foundationsof ModernPhysicalScience(New
York, I926; rev. ed. London, I932).
16 A. Koyre, From the ClosedWorldto the Infinite Universe(Baltimore, I957).
Cited below as "Koyre'."
17 See Koyre, chs. vi, viii.
18 The Scienceof Mechanics,
pp. 271 if.
19 Koyre, p. i6o.

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NEWTON ON ABSOLUTE SPACE

Newton-Logos: this can be shown if we reconsider the -current
doctrine about absolute space. So in what follows I shall first
isolate three main strands in this received doctrine and then go on
to compare them with Newton's own formal pronouncements
on the subject of absolute and relative space. Neither the views
current today nor Leibniz's and Mach's earlier criticisms (I shall
conclude) go to the heart of Newton's distinction. Far from
anticipating Einstein's theory of relativity, Leibniz's attack
on Newtonian space never came to grips with the astrophysical
problems Newton actually solved. As for Koyre's identification
of Newton's doctrines with Henry More's, it is by their effective
scientific fruits that these doctrines must primarily be assessed,
rather than by their supposed metaphysical foundations. These
fruits have, since Mach, been underestimated; and whatever the
historical origins of Newton's ideas, however suggestive or
sympathetic he may have found Henry More's theological
bizarreries, it is far too strong-if not indeed a sheer categorymistake-to say that "Newton's physics ... stands or fall with"20
the substantial or spiritual Space of the seventeenth-century
Neoplatonist theologians.
The received interpretation of Newton's views on space, time,
and motion has, then, three chief strands which need to be
considered separately.
(i) According to the first proposition, time, space, place, and
motion should not (on Newton's account) be referred to any
material objects of reference, but are to be measured "absolutely."
About the positive content of this proposition there are several
views. Newton can be taken as asserting that spatial and temporal magnitudes are intrinsic qualities rather than relations,21
to be measured independently of any reference-system. Mach
adopts this interpretation and understandably objects to it as
vacuous.
A motion is termed uniform in which equal increments of space
described correspond to equal increments of space described by some
Koyre, p. i6o.
Cf. C. D. Broad, "Leibniz's Last Controversy with the Newtonians,"
in Ethicsand the Historyof Philosophy(London, I 952), pp. I68-I9I.
20
21

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motion with which we form a comparison, as the rotation of the earth.
A motion may, with respect to another motion, be uniform. But the
question whether a motion is in itself uniform, is senseless. With just
as little justice, also, may we speak of an "absolute time"-of a time
independent
of change. This absolute time can be measured by comparison with no motion; it has therefore neither a practical nor a
scientific value; and no one is justified in saying that he knows aught
about it. It is an idle metaphysical conception.22
Alternatively, Newton can be taken as believing in the existence
of a fundamental frame of reference in nature, something not
consisting of material objects but having nevertheless an "objective
existence"23 of a nonmaterial or immaterial character, this
frame of reference being either unobservable in principle-a
possibility to which Newton's own words lend some authority24or identified with an all-permeating ether. This last alternative
was much favored around the turn of the twentieth century and is
still occasionally to be met with; from it there arises the idea that
Einstein's principle of relativity, in disposing of the "luminiferous
ether," thereby drove the last nails into the coffin of Newton's
"absolute space."25 But whatever positive meaning is read into
this first proposition, it is always taken to be denying the same
thing: namely, that time, place, space, and motion can, fundamentally, be referred to any material frame of reference. "Curiously enough," says Koyre, "the Cartesian conception of the
only relative, or relational, character of these and connected
notions is branded by Newton as being 'vulgar' and based upon
'prejudices.' "26
22
23
24

The Scienceof Mechanics,pp. 273-274.
Cf. H. Poincare, Scienceand Hypothesis,ch. vii (London, 1905), p. II 7.
"But because the parts of space cannot be seen, or distinguished from one

another by our senses

. . . ,"

Princ., p. 8.

25

See, most recently, M. Jammer, Concepts
of Space(Harvard, 1954), p. 171:
"Thus Newton's conception of absolute space and its equivalent, Lorentz's
ether, were shorn of their ability to define a reference system for the measurement of velocities. This was accomplished by the special theory of relativity."
Sir Edmund Whittaker, on the other hand, makes it clear in his Historyof the
Theoriesof Aetherand Electricity,II (Edinburgh, 1953), 27-28, that the question
of the luminiferous ether only became identified with that of absolute space
during the nineteenth century.
26 Koyre, p. I 6o.
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NEWTON ON ABSOLUTE SPACE

(ii) The second proposition in the received doctrine concerns
the reality of Newton's "absolute," nonmaterial referencesystem. Henry More and Joseph Raphson both insisted, for
theological reasons, on the "real existence" of Space as distinct
from all material things. "Indeed," writes More, "we are unable
not to conceive that a certain immobile extension pervading
everything in infinity has always existed and will exist in all
eternity . . . really distinct from matter."27 Poincar6, again,
takes it for granted that absolute space must be something having
an "objective existence," and textbooks of physics to this day
refer to the notion as that of a "cosmic substratum."28 Their
authors, it is true, do not know whether to apologize for or
congratulate Newton on the use he makes of the notion: on the
one hand, says Joos, absolute space and absolute time are clearly
unnecessary fictions, yet on the other hand
the fact that Newton was able to formulate a thoroughly adequate
foundation for the mechanics of macroscopic bodies by introducing
these two concepts, whose difficulties could scarcely have escaped his
notice, is to be regarded as an outstanding stroke of genius.29
Still, it would be odd if the only way of constructing a satisfactory theory of dynamics had ever been to make a pair of
assumptions as gratuitous and nonfunctional. So perhaps our
doubts should be less about Newton's theory than about this
way of interpreting it.
(iii) The third element in the received doctrine calling for
comment has to do with the historical background of relativity
theory. According to this, Newton's contemporaries-notably
Berkeley, Huygens, and Leibniz-saw at once that the soundness
27

Enchiridiummetaphysicum
(London, i67I), p. 68; quoted in Koyre, p. I46.
G. Joos, TheoreticalPhysics (London, I934), p. 2i6.
29Joos, op. cit., p. 2I7. Notice that the whole sentence is italicized by the
author, and cf. Jammer (op. cit., p. I7I) who writes, "Space as conceived by
Newton proved to be an illusion, although for practical purposes a very
fruitful illusion-indeed, so fruitful that the concepts of absolute space and
absolute time will ever remain the background of our daily experience."
This last remark increases the confusion, since it construes not only Newtonian
dynamics but our everyday nontheoretical ideas also as involving a "metaphysical illusion."
28

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STEPHEN TO ULMIN

of his theory depended on that of its "metaphysicalfoundations."
These foundations, they argued, were essentially defective, and
the objections they put forward on metaphysical and theological
grounds foreshadowed those which Einstein was in due course
to press home, when the time came for him to dethrone Newton's
theory within physics itself.30
II
How far do these three propositions fairly represent Newton's
conception of absolute space? Enough support can be found in
his writings to account for their currency, especially if we cast
our net wide and bring in as evidence the more general and
informal remarks of his later life, as well as passages which were
frankly theological. Yet we should be on our guard and examine
not only Newton's words but also his deeds; the use, that is, that
he makes of the notion of absolute space in the dynamical system
of the Principia.And we should beware of taking too much at
their face value the things a man says either about his method
(knowing "the difference-in science-between what men say
and what men do"31)or about the theological implications of his
theories. We can judge the received doctrine truly only if we
concentrate our attention on the key passage near the beginning
of the Principia where the distinction between absolute and
relative space is first introduced, and-bearing in mind the
scientific problems Newton was tackling-see whether this
passage happily bears the interpretation commonly placed on
it.32 In my view, none of the three propositions I have stated will
stand up to a scrutiny of this kind; and an alternative, happier
reading of Newton's famous scholium can be given, which fits
much better his explanatory practice and must take its place in
any fair and comprehensive account of his views.
The scholium on space, time, and motion follows immediately
after eight "definitions":33 apart from these, it is the first thing
30 Koyr , chs. x and xi, pp. 22I-272;Jammer,

op.cit., pp. I I I-I24.
I. B. Cohen, FranklinandNewton(Philadelphia, I956), p. 47.
The passage in question is the scholium, Princ., pp. 6-i2.
33 Prince:
pp. i-6.

31
32

IO

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NEWTON ON ABSOLUTE SPACE

in the treatise, and (what should be noted) precedes the "Axioms,
or Laws of Motion" with which the formal exposition of Newton's
mathematical theory begins.34 In the initial definitions Newton
has been introducing the technical terms of his theory-quantity
of matter and motion, innate, impressed, and centripetal force,
and the various quantities (absolute, accelerative, and motive)
of a centripetal force. These terms are all dynamical terms of art,
even when not of Newton's own coining, so that he is bound to
indicate how they are going to be used within the treatise which
follows. But when he goes on to explain his spatial and temporal
terms, the situation is different: these terms are familiar to all
and there is accordingly no place for formal definitions.
Hitherto I have laid down the definitions of such words as are less
known, and explained the sense in which I would have them understood in the following discourse. I do not define time, space, place, and
motion, as being well known to all.35
Nevertheless the use which he is going to make of these terms has
certain special features, which marks it off from their common
use, so that he must add an explanation:
Only I must observe, that the common people conceive those
qualities under no other notions but from the relation they bear to
sensible objects. And thence arise certain prejudices, for the removing
of which it will be convenient to distinguish them into absolute and
relative, true and apparent, mathematical and common.36
So far the only thing calling for comment is the allusion to
"the common people." This Koyre takes to be scornful: the
"relational character of the Cartesian conception of space and
time is branded by Newton as being 'vulgar' and as based upon
'prejudices.' "37 Yet surely the Descartes scholar is taking
premature offense. Even if it were in Newton's character to be
contemptuous of the populace, this would be an odd context in
which to evince his feelings, and there is in any case no need to
read his words in that sense. For, in the first place, phrases like
34

Princ., pp. I 3-28.

35

Princ., p. 6.
Ibid.
Koyre',p. i6o.

36
37

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"the common people" or "the vulgar" have become much
more derogatory by now than they were in Newton's time:
Berkeley, for example, in a well-known remark about the philosopher "speaking with the vulgar but thinking with the learned"
clearly meant no more than "ordinary, unphilosophical people."38
Nor does Newton in fact say that the ordinary man's use of the
terms is based upon prejudices, but rather that it is liable to give
rise to them: that is, that a man who is familiar only with the
nontechnical use of spatial and temporal notions will be liable
to misunderstand (prejudge) the specialized use of these terms in
dynamics, unless he is warned how this technical use departs from
the one with which he is familiar. Indeed it is for this very
purpose-to mark off the special character of the use he will be
he goes
making of these terms in his dynamical theory-that
on "to distinguish them into absolute and relative, true and
apparent, mathematical and common."
The wording of his distinctions has often been quoted:
I. Absolute, true, and mathematical time, of itself, and from its
own nature, flows equably without relation to anything external,
and by another name is called duration: relative, apparent, and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is
commonly used instead of true time; such as an hour, a day, a month,
a year.
II. Absolute space, in its own nature, without relation to anything
external, remains always similar and immovable. Relative space is
some movable dimension or measure of the absolute spaces; which
our senses determine by its position to bodies; and which is vulgarly
taken for immovable space; such is the dimension of a subterraneous,
an aerial, or celestial space, determined by its position in respect of
the earth. ...39

Commentators on these distinctions and on the rest of the
scholium which follows have tended to fasten on the more obscure
and metaphysical-seeming statements as revealing the drift of
Newton's thought at this stage. Newton talks, it is true, in several
38

George Berkeley, Principlesof HumanKnowledge,par.

5I.

39 Princ., p. 6.

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NEWTON ON ABSOLUTE SPACE

places in the scholium as though space and time could have
parts in just the same way as material things, with the sole
difference that, if we "suppose those parts to be moved out of
their places, and they will be moved . . . out of themselves. For
times and spaces are . .. the places as well of themselves as of all
other things."40 And this has given force to the suggestion
that, for him, absolute space and absolute time were extra,
quasi-material constituents of the universe. But two things should
make us look more closely at the distinctions. In the first place,
Newton himself betrays obvious discomfort at the idioms to
which he finds himself resorting in order to convey his meaning:
"suppose those parts to be moved out of their places, and they
will be moved (if the expressionmay be allowed) out of themselves.
For times and spaces are, as it were, the places as well of themselves
as of all other things." This by itself raises the doubt whether we
should put too much weight on the remarks in question. The
other point is, if possible, even more obvious: it is that for Newton
the contrast between "absolute" and "relative" time, space,
place, and motion is one and the same as that between "mathematical" and "sensible." Our next task is to explore the implications of this identity.
What can be the force of distinguishing, in this context, between
a "mathematical" and a "sensible" use of spatial and temporal
terms? In what respects is Newton's theory so novel that it is
liable-unless
this distinction is constantly borne in mindto create misunderstandings or "prejudices"? In answering these
questions we have two guides: first, our own knowledge of the
history of dynamics and of the logical structure of scientific
theories (both of them subjects which have been much clarified
since Newton's time), and, second, the things Newton himself
says in the Preface to the first edition of the Principia about the
analogous mathematical character of geometry. 41 Taken together, these things focus attention on the chief and deliberate
originality in logical structure which marks off Newton's theory
of dynamics from so many of its predecessors, namely, its elabo40

Princ., p. 8.

41

Princ., p. xvii.

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rately axiomatic character. The layout of the treatise is clearly
modeled on that of Euclid's Elements, and its peculiarly "mathematical" features arise from that fact.42
At this point we are entitled to take advantage of more recent
insights into the nature of axiomatic systems. For in all discussions
of such logical structures one distinction is a commonplace:
that between mathematical variables considered only for their
place in a formal system and the same variables as figuring in
applications of the system.43 Of course Newton would not have
been in a position to state this general distinction in an unambiguous manner; yet there are grounds for thinking that, in effect,
he is concerned to draw just such a distinction here.
To begin with, Newton's "mathematical" terms are characterized entirely by reference to their necessaryproperties, whereas the
corresponding "common" or "sensible" terms are spoken of as
"measures" of spatial and temporal magnitudes, which satisfy
the basic axioms and definitions of the theory only contingently:
"mathematical time, of itself, and from its own nature, flows
equably .. .; common time is some sensible and external (whether
accurate or unequable) measure of duration by the means of
motion."44 The fundamental difference is between "time" so
understood that it flows equably "from its own nature," and
42 The laws of motion are labeled "axioms," and the
whole exposition is in
terms of "lemmas," "corollaries," "theorems," and the like. Cf. I. B. Cohen,
op. cit., p. i28. A deductive
method of exposition was of course no novelty, but
Newton goes out of his way to separate the formal mathematical exposition
of his dynamical system from its application to the phenomena of nature. The
interpretation I give of Newton's scholium in what follows generally agrees
with that given more briefly in E. W. Strong's paper "Newton's 'Mathematical Way,"' Journalof the Historyof Ideas,XII (I95I), 95-I IO, reprinted in
Roots of ScientificThought (1957),
pp. 412-432;
n.b. esp. pp. 422-424.
My
attention was drawn to this admirable discussion only after I had completed
the first version of my present paper. See also Strong's Proceduresand Metaphysics(I 936) for a general criticism of the Burtt view of Newton.
43 See, for instance, Braithwaite, Scientific
Explanation,pp. 27 ff. Braithwaite's
treatment of a mathematical calculus as "a game for one player played
with marks on paper" is, of course, quite un-Newtonian, but the distinction
between mathematical quantities and their physical measures can be drawn
equally, whether one adopts a "realist" or a "formalist" approach to the
philosophy of mathematics.
44 Princ., p. 6.

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NE WTON ON ABSOLUTE SPACE

measures of time which may or may not be equable. Taken
in this sense, the distinction is not an empirical one between two
separate sets of things, one stationary, one mobile, but rather a
logical one: certain questions, it is implied, make sense of any
"sensible" measure of such magnitudes which simply do not
arise about the corresponding "mathematical" terms. We can
ask whether or not the year is getting longer, but of "time"
dynamically and mathematically defined this question cannot
be asked. We can talk likewise of any material thing whatever
its place-so
that no observable (sensible)
moving-changing
frame of reference can be "immovable" by the simple meaning
of the term ("from its own nature"), but "places" themselves,
supposing we can give the term a purely mathematical sense,
cannot be said to "move,' nor strictly speaking to be "stationary"
either. In Newton's theory, then, the fundamental criteria of
uniformity of motion and identity of position are to be mathematical
ones: all such terms are to be grounded logically in the theory and
not conceived "from the relation they bear to [any particular]
sensible objects."45
Newton's theory being axiomatic in structure, some such
distinction was essential, whether or not Newton himself was
crystal clear in his own mind about its logical character. (He
would have been a double genius if he had been.) But he does
more than state the distinction and leave it at that; he goes on
to indicate how we are to recognize in practice measures of
time and spatial frames of reference suitable for the application
of his theory to the problems which interest him, notably to
problems in planetary dynamics. As he sees-and explainsnot all measures of spatial and temporal magnitudes are for his
purposes equivalent or equally applicable. To this extent, some
measures of duration and sensible spaces (or "movable dimensions") are better realizations than others of the mathematical
ideals of his theory. Solar time, for instance, is only roughly
uniform, and already astronomers had taken to correcting it.
Rotating and accelerating axes of reference have likewise to be
ruled out; one must confine oneself to frames of reference which
45

Ibid.
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are for practical purposes "inertial frames," if the laws of motion
are to be applicable in their simplest form.46
The case of absolute time is instructive, on account of Mach's
complaint that this is "an idle metaphysical conception" which
has "neither a practical nor a scientific value."47 For the illustration Newton offers of its application is both practical and
scientific:
Absolute time, in astronomy, is distinguished from relative, by the
equation or correction of the vulgar time. For natural days are truly
unequal, though they are commonly considered as equal and used
for a measure of time: astronomers correct this inequality for their
more accurate deducing of the celestial motions. ... The necessity
of this equation, for determining the times of a phenomenon, is
evinced as well from the experiments of the pendulum clock, as by the
eclipses of the satellites of Jupiter.48
The allusions in this last sentence are worth a word of explanation.
Suppose we make a pendulum clock as accurately as we can, and
compare the rate of swing of the pendulum with the motion of
the sun across the sky; we shall find that they do not proceed
exactly in step, their relative rates varying between the equinoxes
and the solstices. What are we to make of this discovery? If we
use the "natural" or solar day as our measure of "vulgar time,"
we shall be obliged to say that the pendulum swings at a rate
which varies slightly according to the season of the year, and that
this variation persists however much we improve the construction of the pendulum. (This by itself might be a genuine phenomenon calling for a physical explanation.) But now suppose in
addition that we carefully observe the motions of Jupiter's
satellites,49 timing the instants at which they are eclipsed once
46 The most satisfactory analysis I know of this aspect of the Newtonian
theory is to be found in W. H. Macaulay's paper, "Newton's Theory of
363Kinetics," Bulletin of the AmericanMathematicalSociety,III (i896-i897),
Macaulay uses the phrase "kinetic axes," instead of the more familiar
37I.
"inertial system" introduced by Lange in i885; cf. Jammer, op. cit., pp. I38-

'39.

p.

The Scienceof Mechanics,p.

47
48

Princ., pp. 7-8.

49

What Newton calls elsewhere the "circumjovial planets"; cf. Princ.,

274.

401.

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NEWTON ON ABSOLUTE SPACE

again by solar time: in that case, we shall find a precisely similar
seasonal variation. Yet who will want to concede that the rate
of circulation of these satellites around Jupiter is affected by the
season of the year-the season, that is, not of the jovial but of the
terrestrial year? We clearly do better to regard the pendulum and
the circumjovial satellites alike as moving more uniformly than
the sun across the sky, hoping in due course to explain this
variation in the sun's apparent motion as the more genuine
phenomenon. (An explanation is in fact not hard to find: the
effect is produced by the combination of the earth's orbital
motion with its rotation.) If we accept this alternative, however,
that will mean abandoning the previous "vulgar" measure of
time and taking in its place for theoretical purposes a dynamical
ideal: namely, the common ideal toward which both terrestrial
and satellites-are found to
and celestial motions-pendulums
approximate. In these so nearly uniform motions we see the
closest physical realization of the "absolute, true and mathematical time" of Newton's axiomatic system.50
Newton's model of a mathematical system is geometry, and
there too a similar distinction can be drawn. Straight lines and
circles described in chalk on blackboards, cut out of sheet iron or
painted on walls, satisfy only approximately the specifications laid
down in an axiomatic geometry such as that of Euclid. The
mathematician's ideal straight lines and circles possess "in their
own nature," as Newton might say, characteristics which
"sensible" straight lines and circles possess only more or less in
fact. So one could preface an axiomatic treatise on geometry
with an explanatory scholium doing for terms like "point,"
"line," and "circle" what Newton's own scholium does for
"time," "place," and "motion." This scholium would indicate
how the nonmathematical use of geometrical terms differs from
the specialized, mathematical use, and how "certain prejudices"
might again arise from failing to grasp these differences clearly.
Indeed, Newton does himself refer to this distinction some ten
pages before the disputed scholium on space and time, in the
50 On this subject, W. H. Macaulay is again most penetrating: see his
Britannica,ioth ed., XXXI,
article, "Motion, Laws of," in the Encyclopaedia
7-II.

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preface to the Principia. There he underlines the contrasts between
the mechanical production of geometrical figures and the rational,
demonstrative discussion of their properties. The business of
mathematics is solely with the latter, and it applies to actual
material things only by way of techniques of measurement and
description:
The description of right lines and circles, upon which geometry is
founded, belongs to mechanics. Geometry does not teach us to draw
these lines, but requires them to be drawn; . . . then it shows how by
these operations problems may be solved. To describe right lines and
circles are problems, but not geometrical problems. The solution of
these problems is required from mechanics, and by geometry the use
of them, when so solved, is shown; and it is the glory of geometry
that from those few principles, fetched from without, it is able to
produce so many things. Therefore geometry is founded in mechanical
practice, and is nothing but that part of universal mechanics which
accurately proposes and demonstrates the art of measuring.51
Any satisfactory reading of the contentious scholium on space
and time should take this earlier passage into account. In each
case, Newton contrasts mathematical arguments and ideals with
practical instances or measures. In dynamics as in geometry it
can always be asked how accurately in fact particular figures or
measures realize our ideals, whether of "right lines and circles"
or of "absolute, true" time and space. Sensible standards and
measures of position, velocity, and lapse of time will possess more
or less in fact the characteristics which the variables x, y, z, 4
and their derivatives possess in the theory from their own nature;
and any material frame of reference we choose will be a more or
less good approximation to the ideal of an inertial frame. Not for
nothing was the seventeenth-century revolution in mathematical
physics a Platonizing one,52 for, on this reading, Newton is
echoing Plato's own repeated insistence that theoretical entities
51Princ., p. xvii.
52 See, for instance, Ernst Cassirer, "Galileo's Platonism," in Studiesand
Essays in the Historyof Scienceand LearningOfferedto GeorgeSarton(New York,
pp. 277 ff. and A. Koyre, "Galileo and Plato," Journal of the History
1945),
reprinted in Roots of ScientificThought(1957),
400-428,
of Ideas, IV (I943),

pp.

I47-I75.

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NEWTON ON ABSOLUTE SPACE

be exactly of this
Ideas and the Mathematika-can
only-the
kind or that (roviro), while material things must always be more

or less of one sort or another

(To Troto--ov).53

A general word can be said at this point about the function
of the definitions and scholium which open the Principia. These
have to introduce the reader to terms which are to figure in the
subsequent axiomatic theory: they must accordingly indicate the
sorts of procedure by which the magnitudes defined are to be
measured in practice, and to that extent they will inevitably be
"operational" definitions rather than formal or lexicographical
example by
ones. This has not always been recognized-for
Mach, who objects that Newton's very first definition (of mass)
is circular:
The formulation ... which defines mass to be the quantity of matter
of a body as measured by the product of its volume and density, is
unfortunate. As we can only define density as the mass of unit of
volume, the circle is manifest.54
If we were concerned with a formal definition within an axiomsystem, Mach's objection would carry weight; and once we have
accepted the Newtonian theory in toto as a basis for our thought
about matter and motion, we shall no doubt come to define
(or redefine) density in terms of the more fundamental conception,
mass. But the reader with whom Newton is concerned is not yet
converted; he has opened the treatise without prior knowledge
of the contents; and he has not yet been introduced even to the
laws of motion. That being so, he is in no position to understand
the term "density" as Mach does.55 In fact, Newton's contemporaries must have read the word in an unsophisticated way,
as roughly equivalent to our own "specific gravity"-this being
the "vulgar" sense of the word. And so Mach's circle would have
been broken, and the reader introduced to the idea of mass
operationally, by way of more familiar and easily measured
53Timaeus, 49d.
54 Op. Cit.,

pp. 237-238.

55 I. B. Cohen understands "density" here as "the number of corpuscles

per unit of volume"; but one need hardly bring in Newton's atomism at this
point, any more than one need his subsequent redefinition of density in Book
III (Princ., p. 4I4): see Cohen, op. cit., p. II5 n.
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quantities. The quantity of matter or mass of a body, says Newton
in further elucidation, "is known by the weight of each body;
for it is proportional to the weight, as I have found by experiments
on pendulums, very accurately made, which shall be shown
hereafter."56 It is, therefore, no accident that five of his eight
definitions take the form, "So-and-so is the measureof the same,
arising from (or proportional to) Such-and-such."57

III
In this examination of Newton's scholium on space and time
I have focused attention on the scientific function of Newton's
distinctions. By their "scientific function" is here meant the part
they have to play in the dynamical explanations which Newton's
theory makes possible-for instance, those which the great man
himself gives later in the Principia, notably in Book III where he
applies his theory to "The System of the World."58 Discovering
this function is not a purely historical task. In fact, it requires
us to restrain for the moment our very proper interest in historical
questions of two kinds: first, those about the background from
which Newton's ideas sprang, and, second, those about the wider
implications Newton himself might have wanted to read into his
distinctions. There will be time in due course to ask whether
everything in the passage we are studying can be given a purely
dynamical interpretation, or whether Newton himself did not
read more into his doctrines than we have here allowed. Still,
as a principle of method, it seems to me that this time should be
deferred. So long as a dynamical interpretation remains open to
us we should accept it, just because of what it is, in preference to a
metaphysical or theological one. We may perhaps, on further
study of the scholium, be driven to conclude that at a certain
point Newton leaves physics for metaphysics-that not everything
he says there can be understood in a dynamical sense; but in
Princ., p. i.
Definitions I, II, VI, VII, and VIII. E. W. Strong makes this
point in the paper referred to above (note 42): cf. Rootsof ScientificThought,
56

57 Viz.,

P. 42I.
58

Princ., pp. 397-542.
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reading so austerely restricted and carefully phrased a treatise,
our initial presumption must surely be against this.59
With this distinction in mind, let us return to the received
account of Newton's views on space. If there is any justice in
the present reading of his scholium, that account will have to be
reconsidered. For there is a clear distinction in function between
two uses of spatial and temporal terms, mathematical and common, corresponding to Newton's "absolute" and "relative" space
and time; and this distinction, regarded as a logical distinction,
has few of the implications normally read into it.
To begin with, does Newton say that the only true measures
of spatial and temporal magnitudes are relative to nothing-at-all?
Does he, that is, deny the relational character of these measures?
Not at all. Of course any practical measurement of these
quantities must be related to some materially-identifiable frame
of reference, and so be a more or less perfect realization of the
theoretical ideal in question (that is, more or less accurately
substitutable for the variable in question); but measures of any
kind belong on the "relative, apparent, and common" side of
Newton's dividing-line. Absolute space and time are, on this
reading, not the most perfect measures one could conceive, but
rather the theoretical ideals to which all relative measures are
more or less good approximations.60
From this it follows that no natural reference-frame, however
perfect, could properly be identified with Newton's "absolute
space." Even supposing there were an ether, whether responsible
for the action of gravitational attraction as Newton supposed or
luminiferous on the nineteenth-century model; even this would
59The contrast between Newton's Principiaand his Opticksis well brought
out by I. B. Cohen, op. cit., esp. pp. I25 if. The speculative freedom of the
"queries" at the end of the Opticksis to be found nowhere in the body of the
Principia.
60 This statement needs some qualification. In any Platonizing theory
there is liable to be a certain obscurity about the distinction between mathematical concepts (or Ideas) and their perfect realizations (the Mathematika)between "the circle" and an actual, supposedly-perfect circle. This ambiguity
is undoubtedly present in Newton's scholium, in particular where he talks
of "the true motions" of bodies (Princ., p. io) meaning not notions mathematically conceived, as his definitions would suggest, but motions as known
by some supposedly-perfect measure.
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be, logically speaking, only one more possible reference-frame
providing measures of position, velocity, and the like. It could
never, without a category-mistake, be said to be absolute space.61
Newton's own ether was certainly quite distinct from space; it
was ''only a very thin and very elastic substance, a kind of
exceedingly rare gas . . . composed of exceedingly small particles
between which, of course, there is vacuum." 62And the same trouble afflicts any interpretation of "absolute space" as a "cosmic
substratum" or special kind of quasi-material body. For if one
does imagine some sort of cosmic substratum being found to exist,
so that wherever scientists turned their instruments they observed
a fine rectangular network of lines moving relative to the objects
were this
of the world, but not otherwise affecting them-even
to be discovered, we should not be forced to identify the network
as "absolute space." The question would still arise, as it must of
any material reference-body such as the sun, how far that network
could count as defining an inertial frame of reference. This too
would be at best one more approximation, though perhaps as
perfect a one as our instruments could detect, to the theoretical
ideal Newton calls "absolute space."
Likewise for absolute time. One can imagine that scientists, on
tuning in their radios to a hitherto-unexplored wave length,
might hear ticking going on which could be traced to no terrestrial
transmitter and continued without diminution or interruption
wherever we listened for it. Some people might eventually
conclude that we had thereby discovered a way of listening to
the Deity's own chronometer; yet even so we should not be
obliged to say, "So this is absolute time!" God would presumably
find the Ptolemaic system no more laborious to operate with than
the Copernican, so there is no knowing what sort of time-scale
he might find natural; and the question can accordingly be
raised of the Divine clock, as much as of the solar day, how far it
provides a satisfactory measure of time for the purposes of our
dynamics. Far, therefore, from Newton's absolute space and
absolute time needing to be understood in terms of mysterymedia and -processes, it would have been inconsistent of him to
61
62

Cf. Jammer, op. cit., p.
Koyre, pp. I 7 I - I 72.

I 7.I

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NEWTON ON ABSOLUTE SPACE

use them in this way, since that would have meant going back
on his own distinction between the mathematical and empirical
aspects of spatial and temporal variables.63
The problem of absolute space, which is so often treated as
though it were a metaphysical one, has in fact been bedeviled by
the injection of metaphysical issues at this point. Both Newton's
disciples and his critics have been free in their use of the blanketnouns "Space" and "Time," and have used them with little
circumspection: they have, in particular, allowed themselves to
become preoccupied with one question whose consequences have
been only obfuscating-the question whether absolute space and
absolute time really exist. Newton, as we have seen, is more
cautious and shows signs of discomfort when he approaches the
brink of nonsense. His pages are dotted with verbal dangersignals like "as it were," and these we ignore at our own risk;
he is not to blame for the results if we insist on taking literally
something offered solely as a fafon de parler. Even his remark
about God perceiving things directly in space "as it were in his
sensorium"64 one which roused Leibniz's especial scorn65
is qualified in this way, as Dr. Johnson hastened to point out to
the Rev. Hector Maclean, when the Minister of Col repeated
Leibniz's criticisms in a Hebridean conversation.66
It is often claimed nowadays that Newton's reticence about
metaphysical issues is the result of secretiveness rather than
caution; that he was attempting to conceal his metaphysical
commitments rather than to avoid metaphysical entanglements.67
Yet once again, if there is no dynamical occasion for introducing
the question whether space "exists," why are we obliged to drag
it in? In fact there are good reasons for keeping the question
63

C. Neumann's famous body Alpha, and the sphere of the fixed stars,
which Jammer discusses as possible candidates for defining "absolute space"
(op. cit., pp. IOI-I02)
are, logically speaking, in no better position than any
other material objects; though in practice the night sky does give us the means
building up a good approximation to an "inertial system."
64 Opticks,
Qu. 28.
65 See the opening paper of The Leibniz-Clarke
Correspondence,
par. 3, ed.
H. G. Alexander (Manchester, I956), p. ii. Cited below as L-C.C.
66 See Boswell's Tourto the Hebrides,under
Tuesday, Oct. 5, I773.
67 See, for instance, Koyre, pp. I59-I60.

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out, as the parallel with geometry will make clear. For one need
no more assert that Newton was committed by his theory of
dynamics to the objective existence of a cosmic substratum
called "absolute space" than one need say that a geometer, in
formulating his axiom-system, is committed to the quasi-material
existence of an invisible network of geometrical entities interpenetrating the material world, which alone are for him the
true straight lines and circles. In either case the question of
"objective existence" is a red herring. What matters for Newton's
dynamics is that his theory-which
includes, of course, the
distinction between inertial frames and others-should
have a
physical application. In this sense the question of absolute space
is not one for hypothesis,since it concerns a mathematical notion
whose legitimacy is established as soon as one inertial frame has
been identified, not an unobservable quasi-material medium
brought into the theory to explain things which would otherwise
have been inexplicable.
Notice how the question of "real existence" can confuse the
issue. Some people, after studying a little geometry, begin to say
that there are really no (objectively-existing) straight lines and
circles. No "sensible" straight line or circle, they see, can possess
to an infinite degree of accuracy the properties of Euclid's
mathematical ideals. But to say this, however understandable it
may be, leads only to worse difficulties; for if no straight lines
or circles really exist (one may ask), how can geometry have any
application to the world?68 Once one admits the question of
"objective existence," trouble is bound to ensue, since either to
assert or to deny that geometrical ideals "objectively exist" leads
one equally into paradox. This is just what Poincare found in his
own study of the notion of absolute space.69 To accept the notion,
68 It can be argued that at this point the traditional problem of "universals'"
takes its start; and over universals it is certainly notorious how far the issues
are bedeviled by the question whether, where, and in what form they "exist'"
or "subsist."
69 Scienceand Hypothesis,
ch. vii. When considering at what time the question
of space's real existencebecame important for mathematicians, notice particularly the change in Euler's views remarked on byjammer (op. cit., pp. I27-I 28)
between I 736, when "the question of the real existence of absolute space is a
matter of indifference" to him, and I 750, when he calls it a "chose reelle, qui

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NEWTON ON ABSOLUTE SPACE

he supposed, necessarily involved believing in its "real existence,"
which he could not bring himself to do; even "the absolute
orientation of the universe in space," he declared, "does not
exist."70 All the same, he pondered, rotation does pose problems:
"If there is no absolute space, can a thing turn without turning
with respect to something? . .. All possible solutions are equally
unpleasant to us."'71 And he goes on to present his well-known
fable, about the men for whom the sky had always been covered
with thick cloud, who might yet (he supposes) be led to conjecture
though paradoxically-that
the earth turned round.

IV
I question, therefore, whether the objectiveexistenceof absolute
space was or need have been the central issue for Newton.
Its "reality" or applicability, as a concept, is a different matter.
The need to make use of the concept-to treat spatial magnitudes
mathematically in one's dynamical theories-that
is sufficient
demonstration that the notion is respectable, and not (in Mach's
phrase) a "conceptual monstrosity" or Begrifsungetam.72 As
Clarke puts it, in his fourth reply to Leibniz, "The reality of
space is not a supposition, but is proved by the foregoing arguments; . . . showing, from real effects, that there may be real
motion where there is none relative; and relative motion where
there is none real." And these arguments, he adds, are "not to be
answered, by barely asserting the contrary." 73 It is in this
respect-in taking Newton to be concerned not with the objective
existence of a hypothetical entity called Absolute Space so much
as with the applicability to the world of nature of two kinds of
spatial and temporal concept, which he labels "absolute" and
subsiste hors de notre imagination." The issue is, of course, liable to be confused by the fact that withinmathematics
there is another perfectly respectable
idea of existence-such that two positive roots of the equation x2 6x + 8 = o
exist, but no rational square-root of 2 does. Over existence at this level, however, none of Poincare's problems need arise.
70 Op. cit. (London, I905),
pp. I2I-I22.
71 Op. cit., p.
I I4
72 Mach, op. cit., pp. xxiii-xxiv.
73 L-C.C., Clarke's fourth reply, par. I3, p. 48.

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STEPHEN TO ULMIN

I chiefly contest the received interpretation of
"relative"-that
the scholium on space, time, and motion.
What then are we to make of Newton's famous "bucket experiment"? Here the force of the suggested interpretation comes out
clearly. The passage in question comes toward the end of the
scholium, and is intended to illustrate "the forces of receding
from the axis of circular motion" which, Newton tells us, are
"the effects which distinguish absolute from relative motion."
For [he goes on] there are no such forces in a circular motion purely
relative, but in a true and absolute circular motion, they are greater or
less, according to the quantity of the motion. If a vessel, hung by a
long cord, is so often turned about that the cord is strongly twisted,
then filled with water, and held at rest together with the water;
thereupon, by the sudden action of another force, it is whirled about
the contrary way, and while the cord is untwisting itself, the vessel
continues for some time in this motion; the surface of the water will at
first be plain, as before the vessel began to move; but after that, the
vessel, by gradually communicating its motion to the water, will make
it begin sensibly to revolve, and recede little by little from the middle,
and ascend to the side of the vessel, forming itself into a concave figure
(as I have experienced), and the swifter the motion becomes, the
higher will the water rise, till at last, performing its revolutions in the
same times with the vessel, it becomes relatively at rest in it.74
Notice that this passage does not describe the detailed procedure
of any one actual experiment. It is intended to do just as much
as Newton says and no more, namely, to illustrate the distinction
between absolute and relative motion:
There is only one real circular motion of any one revolving body,
corresponding to only one power of endeavouring to recede from its
axis of motion, and its proper and adequate effect; but relative motions,
in one and the same body are innumerable, according to the various
relations it bears to external bodies, and, like other relations, are
altogether destitute of any real effect, any otherwise than they may
perhaps partake of that one only true motion.75
For Newton's purpose to recount the precise conduct and results
Princ., p.
75Princ., p.
74

I0.
II.

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NEWTON ON ABSOLUTE SPACE

of an actual experiments would have been beside the point;
all that was required was an indication of the sort of familiar
phenomenon in connection with which the point of his distinction
should be obvious.
The point should be obvious, I say; but to many of Newton's
modern critics it does not seem to have been obvious at all.
Following Mach, they have understood him to be using the
phenomenon as evidence of the objective existence of a bogus
entity, whereas nothing in the passage itself claims more than the
dynamical indispensability of a conceptual distinction. What is the
origin of this ambiguity? It arises, I suspect, from attempts to
state the moral of the bucket effect too concisely. The nature
of the effect itself is stated easily enough: that the surface of the
water in a bucket may be eitherplane or more or less concave
(through the action of centrifugal forces) even though the angular
velocity of the bucket relative to the water within it is the same
in every case. If all frames of reference were indifferent for
dynamical purposes, this possibility could not be admitted, since a
given relative rotation should lead to one and only one value for
the centrifugal forces, and so for the degree of concavity. Hence
(Newton argues) not all reference-objects are on a footing, and one
can distinguish rotation as "absolute" only by reference to a
mathematical ideal.
But surely, it may be said, Newton is inferring something much
simpler, namely, that "there is (such a thing as) Absolute Space."
At this point ambiguity sets in. For this form of words maymean
only "The concepts of absolute rotation and acceleration have
application," which is clear and true; but it is more commonly
read as meaning, more obscurely and dubiously, "Absolute
Space exists." This formulation, too, mightbe clear and unexceptionable, if it meant "Inertial frames of reference can be
actually identified in Nature"; but again an alternative translation is usually offered: "Some absolute reference-object exists
in Nature." Even this interpretation couldbe rescued if one read
it as meaning "Objects can in fact be found which satisfy for
76 As he does, for instance, in the General Scholium to Prop. XXXI of
Book II, where he describes his experiments with pendulums having bobs of
different materials: see Princ., pp. 3i6-326.

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STEPHEN TO ULMIN

practical purposes the dynamical specification for absolute
reference-objects," but again Mach and his followers have
which can more easily be
adopted a different reading-one
motion relative to certain
means
motion
ridiculed: "Absolute
objects which are absolutely stationary by definition, viz., that
invisible continuum of quasi-material points which constitute
Absolute Space." Mach of course insists that no inference to the
objective existence of an Absolute Space is justified; if the idea
of absolute motion is to have a sense, it must meanmotion relative
to the fixed stars, and it is these stars, not an invisible continuum
of quasi-material points, which constitute the basic referenceobjects for dynamical theory.77
Newton's actual meaning, in my view, is one stage more
sophisticated than Mach recognizes. No material object can, on
Newton's account, be absolutely stationary by definition; the
bucket experiment proves only the practicalrelevance of the distinction between absolute and relative rotation. This fact is not a
truth of logic but a phenomenon of physics. It calls not for a
definition but for an explanation, and merely to state it implies no
reference to the fixed stars or to anything else. Of course, one is at
liberty to try and explain this phenomenon (the inertial properties
of matter) by invoking hypotheses about the influence of the
fixed stars. Einstein and Sciama have in fact done this, arguing
that the inertial properties of material objects in one part of the
universe are the outcome of interactions with all the rest of the
matter in the universe, including the most distant stars. If one
does this, the fact that the directions of the fixed stars can themselves be used to determine the axes of an inertial frame will no
doubt cease to be so mysterious. But however much the stars may
enter into one's physical explanation of inertial effects, they
remain in logic as unfitted as any other physical objects to enter
into the meaning of the term "absolute rotation."
The one clear inference from the bucket-effect is, accordingly,
that we must distinguish between inertial frames of reference,
rotation with respect to which can count as absolute rotation, and
77Cf. Mach, op. cit., 4th ed., p. 543: he has no objection to the phrase
"absolute" rotation, "if it be remembered that nothing is meant by such a
designation except relative rotation with respect to the fixed stars."
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NEWTON ON ABSOLUTE SPACE

other frames. Only if all frames of reference and measure of
spatial magnitudes were proved to be dynamically as good as one
another might it be argued that the notion of absolute, mathematical space could now be entirely dispensed with. This, on the
level of physics, not metaphysics, is a state of affairs Einstein spent
much of his professional career trying to bring about. But whatever his two theories of relativity achieved, the logical distinction
between mathematical variables and sensible measures still
remains; and there is also still room, even within relativistic
dynamics, for the distinction between inertial and other frames
of reference.78 "Thus, if 'absolute rotation' is interpreted as rotation relative to an inertial frame, it is still [to this day] as meaningful a phrase as in Newtonian dynamics."79
(To be concluded)
STEPHEN TOULMIN

University of Leeds

78 On this see Whittaker, Historyof the Theoriesof Aetherand Electricity,II,
chs. ii, v, esp. p. i6o.
79 H. G. Alexander,
in L-C.C., Introduction, p. liv.

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