Publsihed in Philosophy of Science 80, 22-48 (2013).1

Rethinking Newton’s Principia

Simon Saunders

Abstract

It is widely accepted that the notion of an inertial frame is central to
Newtonian mechanics and that the correct space-time structure underly-
ing Newtons methods in Principia is neo-Newtonian or Galilean space-
time. I argue to the contrary that inertial frames are not needed in
Newtons theory of motion, and that the right space-time structure for
Newtons Principia requires the notion of parallelism of spatial directions
at different times and nothing more.

1. Introduction

Few writings in physics have occasioned as much philosophical interest as New-
ton’s Principia, but still puzzles remain. One is an alleged inconsistency in
Newton’s stated principles in application to a universe that is, quite possibly
like the actual universe, infinite and approximately homogeneous in space.2 An-
other is that the motions identified by Newton’s methods as privileged (‘true’,
according to Newton) could not, realistically, be considered even as motions
relative to an inertial frame3 – or not if the universe is sufficiently large.

The two problems are evidently related. Indeed, if the material universe is in
fact infinite and if Newton’s theory really is inconsistent in that application we
should expect that no true motions in Newton’s sense could have been identified.
But that raises a further puzzle in its own right. For Newton surely did pick out
definite motions as somehow preferred; his theory as laid down in the Principia
(what I shall call Newton’s theory of motion or NTM) was empirically successful;
it was genuinely informative: How can an inconsistent theory be empirically
informative?

Of course there are other cases of alleged inconsistencies in empirically suc-
cessful theories. Relativistic quantum electrodynamics, with its famous diver-
gences, is another example. It is tempting to place NTM in the same category,
and to view the inconsistency problem as an instance of a more general puzzle

1To contact the author, write to Merton College, Oxford OX1 4JD, or email si-
mon.saunders@merton.ox.ac.uk. My thanks for encouragement, corrections, and criticisms
to Julian Barbour, Harvey Brown, Oliver Pooley, Robert DiSalle, George Smith, and David
Wallace. I owe a particular debt of gratitude to Julian: in conversations and writings he has
long influenced my thinking on space-time matters.

2See e.g. Norton [1993], Vickers [2009]. For the early history of this problem, see Norton
[1999].

3This question was first raised by Kant. I shall come back to Kant’s views shortly.

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(of how inconsistencies can in practise be ‘contained’). That has been the tenor
of several recent discussions of the subject.4

That is not the strategy taken here. My objective is to give a consistent
reading of Newton’s Principia – but largely discarding the Scholium to the
Definitions – under which it applies equally to an infinite as to a finite mass dis-
tribution, and hence (if only in the non-relativistic limit) to the actual universe.
On this reading the nearest thing to an inertial frame is a non-rotating frame
with arbitrary (and possibly time-dependent) linear acceleration. The privileged
frame (the centre of mass of the solar system), wrongly identified by Newton as
inertial, indeed as at rest, is exemplary: it is a local freely-falling non-rotating
frame. The latter, moreover, is just the sort of frame that is privileged in
Einstein’s general theory of relativity (GTR),5 and in Cartan’s non-relativistic
theory of gravity (so-called Newton-Cartan theory or NCT). These theories have
more in common with NTM than is normally thought.

There is more. As John Norton has recently explained (Norton [1995]),
acceleration in NCT in an infinite universe is relative, not absolute. On the
reading of Principia I shall propose the same is true, but in a rather more explicit
sense. It is that accelerations, as all other motions, are relational in roughly
the sense of the absolute-relational debate in Newton’s time.6 This reading
extends to any theory satisfying Newton’s laws, yielding a form of relationalism
that applies to the entire scope of classical particle mechanics and Newtonian
gravity.7 In this respect it is unlike relationalism in the Machian sense, as
developed by Julian Barbour.8 The latter has the following feature: only systems
which (in NTM terms) have total energy and angular momentum zero can be
defined. No such constraint – and no such prediction – follows from relationalism
in our sense.

What about relationalism in Norton’s sense, in NCT? To explain this we will
need to use the modern apparatus of differentiable geometry, and the concept
of a connection Γ – a rule for defining differentiation of vectors by continuous
transport along smooth curves. Thus, suppose there is a manifold M equipped
with global rectilinear coordinates xα = 〈x0,xj〉, j = 1, 2, 3 (so M has the topol-
ogy of a Euclidean space). The coordinates xα are eventually to be identified
as inertial coordinates. In terms of these, let the connection Γ have as its only
non-zero components the quantities:

Γ
j
00 =

∂φ

∂xj
(1)

4An exception is Malament [1995], who showed how the problem can be solved (effectively
by going over to Newton-Cartan theory; see below).

5See Knox [2013, 2014] for a defence of the claim that such frames play the same functional
role in GTR that inertial frames in practise played in NTG.

6Although it is not motivated in the same way.
7I also believe that it extends to non-relativistic field theory, but I will not try to defend that

claim here. See Hood [1970], Rosen [1972] for the construction of such a theory, specifically
in non-relativistic quantum mechanics.

8See Barbour [1999] for a masterly and self-contained introduction.

2



where φ (the gravitational potential) satisfies:

∇2φ = −4πGρ (2)

with G is the universal gravitational constant G = 6.67 × 10−11 m3 sec−2 kg−1
and ρ the mass density. The equation of motion for a point particle with curve
x : R → M, parameterized by real numbers λ, is then the same as in GTR; it
is the geodesic equation:

d2xα

dλ2
= Γαβγ

dxβ

dλ

dxγ

dλ
. (3)

Affine geodesics are curves whose tangent vector dx
β

dλ
at each point is the parallel

transport of the tangent vector at any other point. Curves that depart from
these represent the motions of accelerated bodies.

One can free the theory from dependence on a special coordinate system
by introducing spatial and temporal metrics, satisfying suitable compatability
conditions, and by introducing a Riemann tensor as defined by the connection.
The theory can then be written in a generally covariant form as in GTR.

The result is Newton-Cartan theory. It is more general than the theory
based on (1) through (3). When the total mass is finite (an ‘island universe’)
the connection has a unique decomposition into a gravitational and inertial part:

Γ = Γgrav + Γinertial (4)

such that there exist global rectilinear coordinates in which Γgrav satisfies (1)
and Γinertial vanishes. Acceleration, as defined by departures from straight-line
motions with respect to Γinertial , is then uniquely defined. More prosaically, in
the coordinates in which Γinertial vanishes, non-accelerating motions are given
by linear equations in x0 and xj. In this context (and in this sense) acceleration
is absolute. But Γ can still be defined even in the case of an infinite and
homogeneous mass distribution. In that case the decomposition (4) is no longer
unique, there are no privileged inertial frames, and there is no absolute meaning
to acceleration. Acceleration is only defined relative to a decomposition of the
connection into an inertial and a gravitational part, and in the case of an infinite
mass distribution, no such decomposition is preferred.

When I say NTM can be read as a relational theory, I mean something rather
different. In the first instance we are speaking of a particle theory, not a field
theory. The mathematics is also elementary – it is a reading of Principia. And
my claim is that motions are relative whether or not the matter distribution is
infinite, in roughly the way relationalists like Huygens and Leibniz thought it
was. In the case of a finite number of particles, it is true, the system as a whole
can be taken to define a frame of reference that is inertial as in the usual reading
of Principia. Kant indeed took this as the only true inertial frame obtainable
by Newton’s methods, with the solar system only the first step (yielding only
a first approximation) in an iterative procedure that in principle can only be
completed when the entire universe is taken into account. Mach, and arguably

3



Leibniz, thought the same. I am taking the opposite strategy to Kant’s. I am
suggesting a reading in which this limit is not necessary to the structure of the
theory and plays no role in its application, and in which the concept of inertial
motion in the usual sense may not even be defined.

As emphasized by Michael Friedman [1992] and Robert DiSalle [2006], the
key to applying Newton’s theory of gravity to only a part of the actual mat-
ter distribution – to beginning the iterative procedure envisaged by Kant as
concluding only with the entire universe – is Corollary VI to the Laws. On
my reading of Principia it is put to wider use: Corollary VI frees the the-
ory from the need to give any operational significance to the notion of inertial
frame. And it poses a further question. Newton thought absolute space and
time and absolute velocities were needed to make sense of his Definitions and
Laws, but the latter admitted more symmetries; they included boosts among
inertial frames as proved in Corollary V (the relativity principle). As a result
Newton’s absolute space and time and absolute velocities were dispensed with,
and replaced by Galilean space-time and relative velocities, with accelerations
as the only absolute quantities. But the Definitions and Laws, as applied to the
motions of bodies ‘among themselves’, have still another symmetry, namely the
group of time-dependent boosts among non-rotating frames, as demonstrated
by Corollary VI. What should then replace Galilean space-time and absolute
accelerations?

I begin with the passage from Newtonian to Galilean space-time. This was
consistent with all the propositions of Bk.1 and Bk.2 of Principia, but it did
violence to the Scholium to the Definitions; it is a conservative extension of
NTM all the same. It is our template for what follows. In §3 I shall give an
argument from cosmology – a near neighbour to the one I began with above –
to show we cannot in practise determine any inertial motions, not in an infinite
nor in a finite universe (so long as it is sufficiently large), by Newton’s principles
– so knowing those motions cannot possibly have played any role in the actual
application of NTM to planetary motions, or to any others that have actually
been observed. The section following is on Corollary VI, and the explanation of
the irrelevance of inertial motions to Newton’s model of the solar system even
in the favourable case of an island universe. In §6 and 7 the ladder is kicked
away. The last sections are on relationalism and the definition of the ‘right’
space-time for Principia.

2. Newton’s absolute space

Newton worked hard, in the Scholium to the Definitions, to explain why a theory
of motion required an absolute, immaterial backdrop – absolute space – that
was not subject to motions of its parts or to any change. He saw plainly the
inadequacy of an appeal to the stellar background or any other observable body
as an absolute standard of rest: that could define only a ‘relative space’, a
‘movable dimension or measure’ of motion – quantities that may have nothing

4



to do with true motions. No observable body was immovable. No idealization
was safe.

The avowed aim of Principia, as stated in the Scholium to the Definitions,
was to show how these true motions were to be determined, including absolute
velocities. The Laws that immediately followed spoke of straight-line motions,
and motions deviating from these – absolute accelerations. Non-accelerating
motions were ‘straight’ by reference to the sequence of positions of bodies in
absolute space at equal times, with equal distances traversed in equal (absolute)
times (in 4-dimensional terms, straight lines in Newtonian space-time). These
locations in space, and space itself, were part of the furniture of the universe
even if they were insensible. The relative motions of bodies, subject to forces
satisfying the Laws, were supposed to eventually yield the true motions. But
by wide assent what Newton’s methods actually gave was motions referred to
Galilean rather than Newtonian space-time, in which no velocity is preferred.

Relationalists like Huygens and Leibniz went further. According to Leibniz,
space was ‘nothing at all without bodies, but the possibility of placing them’;
and ‘instants, consider’d without the things, are nothing at all; ... they consist
only in the successive order of things’ (Alexander [1956 p.26-27]). His most
damaging criticism was that according to Newton’s principles, supposedly de-
terminate physical properties and relations were undetermined by any possible
measurement, however indirect – not just absolute positions but the velocities
of particles too. They could be made whatever you like – there could be no
‘sufficient reason’ as to why they are thus and so – nor could you ever know
that they are thus and so. Neither the speed nor the direction of the motion of
the material universe as a whole, with respect to absolute space, could have any
empirical meaning; hence neither could the absolute speed or absolute direction
of motion of any body within it.

The critique applies equally to Galilean (or for that matter to Minkowski)
space-time, if taken in the substantivalist sense (I shall come back to this sense in
a moment), but it is particularly telling in the context of Newton’s absolute space
and time. There the problem can be stated independent of Leibniz’s principles
and of substantivalism. In space-time terms, Newton’s absolute space picks
out a preferred velocity – absolute rest – but the laws of motion do not. The
symmetry group of Newtonian space-time is thus smaller than the symmetry
group of the laws. The theory itself says there are these absolute quantities,
whose values can never be obtained, for from the laws it follows (the principle
of relativity):9

COROLLARY V

The motions of bodies included in a given space are the same among
themselves, whether that space is at rest, or moves uniformly for-
wards in a right line without any circular motion

9I am taking the laws to include the requirement that mass is a scalar and that force-
functions do not depend on absolute positions or velocities, for only then is Corollary V
a corollary (see Barbour [1989 p.31-2], Brown [2005 p.37-8]). From sec. 5 on I shall, for
simplicity, assume force functions do not depend on relative velocities either.

5



Given that the motions of bodies among themselves – if necessary, all bodies –
are all that any possible observation can ever have to go on, absolute velocities
can never be determined.

Newton was clearly aware of this difficulty. For textual evidence, consider:
the third edition of Principia contains only three hypotheses, one in Book 2,
two in Book 3, the first and third of a technical nature. The second reads

HYPOTHESIS I

That the centre of the system of the world is immovable

It was followed with the remark:

This is acknowledged by all, while some contend that the earth,
others that the sun, is fixed in that centre. Let us see what may
from hence follow.

The proposition which followed and its proof resolved Leibniz’s under-determination
problem – in a manner of speaking:

PROPOSITION XI. THEOREM XI

That the common centre of gravity of the earth, the sun, and all the
planets is immovable.
For (by corollary iv of the Laws) that centre either is at rest, or
moves uniformly forwards in a right line; but if that centre moved,
the centre of the world would move also, against the Hypothesis.

From the point of view of Galilean space-time, having argued, by Corollary IV,
that the centre of mass of the solar system moves inertially, Newton took it as
a matter of convention that it be considered at rest. I shall come back to this
use of Corollary IV shortly.

Newton’s appeal to absolute space was excoriated by Mach in his early writ-
ings on mechanics. It was considered by Hans Reichenbach and other members
of the Vienna circle as purely metaphysical (one might have thought logical em-
piricists would look on Newton’s conventionalist maneuvering with more sym-
pathy). But philosophers of science have been more forgiving of Newton since:
Howard Stein, in particular, did much to rehabilitate him.

How does Galilean space-time stand up to Leibniz’s critique? A general ap-
proach to the treatment of exact symmetries is this: insist that physically real
quantities, and by extension, physically real properties and relations built out
of those quantities, as specified by a physical theory, be invariant under the
symmetries of that theory.10 In the Galilean case that lets in relative angles
and distances, (magnitudes of) relative velocities, and (magnitudes of) abso-
lute accelerations; it disallows absolute directions and absolute speeds. The

10I have defended this reading of exact symmetries in physics in several places; see, e.g.
Saunders [2003a,b], [2006], [2013].

6



symmetry group of the space-time is bigger, and now includes the symmetries
of the relativity principle; and the theory (in terms of invariants) no longer
speaks of absolute positions or velocities, but only of (magnitudes of) absolute
accelerations. We then have all the quantities ordinarily thought to be measur-
able in NTM, and get rid of the ones forever underdetermined by any possible
observation.

Commitment to invariant quantities does not in itself rule on the reality
of points of space-time; for example, such points can still be relationally dis-
cerned by invariant quantities (again: relative distances, angles, and intervals
in time). But if space-time points are to bear spatiotemporal relations to ma-
terial particles, as substantivalists assume, the difficulty recurs: such relations
will be changed by symmetry transformations, actively construed – by relative
drags between material bodies and points of space-time – hence they are not
physically real. The same difficulty arises for Minkowski space and for general
relativistic space-times with symmetries.11

So much is familiar ground. We are about to move on to new territory, and
to the question of what Newton’s laws really allow us to determine. But to that
end we will need some of the machinery of affine-space theory, and it will be
helpful to first set it up for the familiar space-times just encountered.12

An affine space has much of the structure of a vector space. It is a set in
which pairs of points define vectors, used in turn to define the straight lines
talked of in the Laws – the straight line motions, real or imagined, of bodies
moving inertially. Quite generally, let X be a set, let Vn be an n-dimensional
vector space, and let + on X be an action of Vn on X which is:

• transitive, that is, for any a,b ∈ X , there is a vector −→c ∈ Vn such that
a + −→c = b.

• free, that is, if for some a ∈X , a + −→c = a +
−→
d , then −→c =

−→
d .

If free, the vector in Vn (which exists by transitivity) is unique, denote a− b.
The structure 〈X ,Vn, +〉 is an (n-dimensional) affine space, denote An. A

notion of parallelism for pairs of points can now be defined: a,b is parallel to
a′,b′’ if a− b = a′ − b′. A transformation g : An → An is affine if it preserves
parallels, i.e. if a− b = a′ − b′, then g(a) −g(b) = g(a′) −g(b′).

If Vn is equipped with the Euclidean scalar product with norm |.|, then a
relative distance function h is defined on An× An by h(a,b) = |a− b|n. Since
An is affine, h is non-degenerate and satisfies the triangle inequality: it is a
metric on An. The resulting space 〈X ,Vn, +,h〉 is n-dimensional Euclidean
space.

11I do not myself think this difficulty is insurmountable, however, for reasons sketched in
my [2003a,b]. See my [2013 Section 3.2] for an analogous difficulty in the case of permutation
symmetry.

12We could use the more usual perspective of a differentiable manifold, but besides keeping
the mathematics as simple as possible, I am concerned to use methods as close as possible as
those of Principia (an ‘affine-space plus’ theory in Stachel’s [1993] terminology).

7



It is now clear how to proceed. Space-time should be a 4−dimensional
affine space (to define straight-line inertial motions), but if it is to respect the
relativity principle its Euclidean structure should be limited to simultaneous
3−dimensional surfaces with no preferred velocity (so no decomposition of vec-
tors in V4 into a spatial and temporal part). Define neo-Newtonian space-time
as the structure 〈X ,V4, +,W,h,t〉, where

• V4 is a 4-dimensional real vector space

• + is a free, transitive action of V4 on X

• W is a 3-dimensional subspace of V4

• h is a Euclidean inner product on W, defining a norm |.|3

• t is a Euclidean inner product on the quotient space V4/W , defining a
norm |.|t.

Since it is a 4-dimensional affine space the notion of straight line, and straight-
line parallelism, is well-defined. The lines which (as vectors) lie in W connect
simultaneous points in X , i.e. points a, b such that a − b ∈ W ; and for such
points h defines a metric h(a,b) = |a− b|n on points in X . We could, indeed,
have begun with a simultaneity relation on X and defined W in its terms.13

The affine transformations that preserve S,t and h are the Galilean transfor-
mations (in which I include mirroring and time reversal). Defining a rectilinear
coordinate system −→x ,t on X so as to assign equal times to simultaneous points,
they are:

−→x → R ·−→x + −→u t +
−→
d (5)

t → t + s,t →−t

where R is a matrix of determinant ±1, −→u ,
−→
d are 3-dimensional Euclidean vec-

tors, and s is a real number. Excluding the inversions, these transformations are
parameterized by 10 real numbers, the (proper, orthochronous, inhomogeneous)
Galilean group. In this way we turn our ’affine-space plus’ into a differentiable
manifold (Galilean space-time). The measurable quantities – the invariants of
these transformations – are time intervals, instantaneous relative distances and
their derivatives, and (norms of) relative velocities and absolute accelerations,
and derivatives.

What then is Newtonian space-time? It is the structure 〈X ,V3⊕V1, +,h,t〉;
that is, it is neo-Newtonian space-time with the additional structure that any
vector in V4 is uniquely decomposed into the sum of a vector in V3 (with the
Euclidean metric h) and a vector in V1 (with the temporal metric t). Then any
two points a,b in X have a unique spatial separation, namely h(a,b) = |P(a−b)|3,
where P is the projector from V3 ⊕V1 to V3.

13The method followed by Stein [1967].

8



Invariance of the decomposition V4=V3⊕V1 narrows the space-time symme-
tries to the 7-dimensional subgroup consisting of rotations in space and trans-
lations in space and time. The relativity principle may be a symmetry of the
laws, but it is not a symmetry of the space-time. Only with neo-Newtonian (or
Galilean) space-time are the two correctly aligned. Or are they?

3. An inconvenient truth14

By the late 19th century, analytical methods took centre stage, and increasingly
group theory. The concept of inertial coordinates was the crucial one, and of
inertial frame. The latter term was first coined by Ludwig Lange in 1885,
taking up a suggestion by Carl Neumann in his habilitation address of 1869.
Neumann’s work, published as ‘On the principles of the Galilean-Newtonian
theory’ in 1870, began the debates over the foundations of NTM by Mach and
others that cleared the way for Einstein’s discoveries.

Neumann’s monograph was followed soon after by Ernst Mach’s History and
Root of the Principle of the Conservation of Energy ; both argued for the need
for operational definitions in mechanics. The following year the mathematician
Peter Tait offered the following construction (Tait [1983]): let there be three
force-free bodies, no two of which are at relative rest, that are non-collinear;
let the sequence of equal increments in the relative distance of any two serve
as a clock; then their relative distances and angles with the third, if inertial,
must satisfy certain constraints over time. In fact the first three instantaneous
relative configurations – ‘snapshots’ – are unconstrained (any will do, inertial or
no); only with the fourth is there a constraint, amounting to a test of whether
the three bodies are really inertial.

Tait’s construction was followed by several others, among them one by Lange.
But its weakness as an operational definition of inertial motion was also clear,
for it depended on the availability of force-free (‘fundamental’) bodies. Which,
precisely, were these? Unlike every other force, gravitational force cannot be
screened off. The proposal, like Newton’s appeal to absolute space, was a purely
conceptual device: it had nothing to do with anything that could in practise be
measured. As Bertrand Russell complained:

If motion means motion relative to fundamental bodies (and if not,
their introduction is no gain from a logical point of view), then
the law of gravitation becomes strictly meaningless if taken to be
universal – a view which seems impossible to defend. The theory
requires that there should be matter not subject to any forces, and
this is denied by the law of gravitation. (Russell [1903].)

The situation is different if we use Newton’s methods. Rather than force-
free, he used gravitating bodies, principally the earth, sun and moon. Because

14The first part of this section draws heavily on Barbour [1989 Ch.12, 1999].

9



of the complexity of this system he made use of an iterative procedure, but
in essence the method was an operational one. Thus to a first approximation
he used the area law (Kepler’s second law), itself derived from his principles
for two bodies subject only to centripetal forces, to define a clock (with equal
areas swept out in equal times). Time defined in this way, taking into account
perturbations introduced by a third (and further bodies, notably jupiter), is
ephemeris time. It is time defined so as to make the observed relative angles
subtended by the planets at the earth satisfy Newton’s equations of motion. In
the early 1950s it had acquired sufficient precision to replace sidereal time as
the standard of time. It was replaced in turn by atomic clock time in 1972.

Does that secure the status of Proposition XI, but replacing ‘at rest’ by ‘in-
ertial’ ? No, for the same problem returns: the solar system as a whole is no
more free from gravitational forces than the ‘fundamental bodies’ of which Rus-
sell spoke. The sun and planets are acted on by bodies beyond the solar system:
in principle, arbitrarily many.15 Newton’s reasoning in deriving Proposition XI,
even granting Hypothesis I, was fallacious. Here is Corollary IV to the Laws,
on which the proof turns:

COROLLARY IV

The common centre of gravity of two or more bodies does not alter its
state of motion or rest by the actions of the bodies among themselves;
and therefore the common centre of gravity of all bodies acting upon
each other (excluding external actions and impediments) is either at
rest, or moves uniformly in a right line.

The proviso ‘excluding external actions’ robs it of any application to local grav-
itating bodies, like the planets and sun, which cannot be screened from the rest
of the universe.

Are the accelerations thus produced at least likely to be small, so that Propo-
sition XI (given Hypothesis I) holds to a good approximation? The acceleration
due to the stars in our local galaxy is certainly tiny – no more than 10−10 msec−2

– but include ever-larger regions of space filled with stars and their influence
builds. It is the same as with Olbers’ paradox: the influence of remote stars falls
off as the inverse square, but the number of stars at a given distance, if uniform,
increases as the square. The result is an acceleration that scales linearly with
the diameter of the universe, given an approximately uniform mass distribution.

Any elementary observation that escaped Newton should be spelt out in
full.16 Suppose, for simplicity, matter is uniformly distributed with density ρ
over an enormously large sphere, and suppose the solar system is located within

15Recognition of this point was slow. The question of the proper motions of stars among
themselves was widely discussed by the mid 19th century, as we learn from Mary Somerville’s
On the Connexion of the Physical Sciences [1840] - but not much before, and not even then
in terms of their action on the solar system. (I am grateful to George Smith on this point).
Nor was the universality of gravity taken for granted: Somerville cited the motions of binary
stars as conforming to Newton’s laws as evidence that the stars, and not just bodies in the
solar system, truly gravitate.

16It was also missed by Stein. Having correctly observed ‘Newton’s theory affords a way

10



the sphere at distance R from its centre. The gravitational forces due to the
matter at distance greater than R cancels, leaving only that of the matter in
the interior. This mass, denote M, is as a function of R:

M(R) =
4

3
πR3ρ.

The gravitational action of the sphere of mass M on the solar system (by hy-
pothesis located on the boundary of the sphere) is the same as if all the mass
were concentrated at the centre, and hence produces an acceleration a (using
the proportionality of gravitational to inertial mass):

a =
M(R)G

R2
=

4

3
πGρR. (6)

The result is not materially effected on removing the simplifying symmetries
(so long as the mass density is roughly uniform). For sufficiently large R, the
acceleration of the solar system will as large as you like – or do not like.

But there is always the centre of mass frame for the universe as a whole,
however large, so long as the total mass is finite. To this Corollary IV applies
exactly and without ambiguity. Relative to this frame, whatever it is, if stellar
densities are similar to those of our local Hubble volume, the acceleration of the
earth due to the rest of the universe swamps that due to the sun for a universe
of size 1038 m and greater. As it happens, that is much larger than the visible
universe (by about twelve orders of magnitude) – but there is every reason to
think the actual universe is much larger, if not in fact infinite.

What if the actual universe is infinite? In that case there is a new difficulty:
not so much that the ‘true’ acceleration of the solar system is enormously large
and unknown, but that it is undefined. This problem was first recognized by
Hugo Seelinger in 1894, and was much discussed subsequently. By the above
method the acceleration of the solar system due to the rest of the universe
can be given any number whatsoever.17 The problem was considered so severe
that some concluded the inverse square law itself should be modified. Shortly
after, Einstein made the same suggestion, but for a very different reason. He
too assumed the universe was approximately homogeneous, but in terms of his
newly discovered theory of GTR; and in that context he saw it could not possibly
be static. The solution was to modify the field equations, thus introducing
his ‘cosmological constant’. But now exactly the same reasoning applied to
Newton’s theory of gravity, which Einstein used as a warming-up exercise. The

of assigning kinematical states up to a Galilean transformation, on condition that one has
succeeded in accounting completely for the relative motions by a system of action-reaction
pairs, and on the further condition that there is no reason to suspect the system in question
to be subject to an outside influence imparting equal accelerations to all its members’, Stein
[1977] continued: ‘Newton had the good luck to find such a system: namely, the solar system;
and the skill to effect its thorough dynamical explication.’ It would be by good luck indeed
were this to be true.

17This was the inconsistency discussed by Norton [1993], [1999].

11



modification to Newton’s theory of gravity eliminated the divergence of Eq.(6)
as R →∞.18

Pursuit of these questions led directly to big-bang cosmologies on the one
hand (because Einstein’s solution was ineffectual), and to Newton-Cartan theory
on the other (because Einstein’s geometric methods were so successful). We saw
in section 1 how the problem of acceleration is resolved in that setting. But let
us not take our eye off the ball. There is not and never was any evidence to show
the universe is less than 1038 m in size. Even in the favourable case, supposing
we live in an island universe, by Newton’s principles we never had good reason
to think the solar system or any other concretely-defined frame of reference was
even approximately inertial. That is, the central concept of NTM, the concept
of an inertial frame of reference, was never in practice operationally defined by
Newton’s principles, not even approximately.

4. Corollary VI

The conclusion just reached has an air of paradox. It seems self-defeating if
true: how is it NTM was, in fact, successfully applied?

In only one place in Principia did Newton raise the question of the influence
of the stars on the solar system: in Corollary II to Proposition IV of Book 3. The
proposition itself was that the aphelions and nodes of the orbits of the planets
are fixed (the aphelion of an orbit is its furthest point from the sun; its nodes are
the points of intersection of the plane of an orbit with the ecliptic). The result
was proved in Book 1 (at Proposition XI) in the approximation in which the
mutual actions of the planets and comets among themselves are neglected, but
Newton stated it again in Book 3 to draw two corollaries. The first is that ‘the
fixed stars are immovable, seeing they keep the same position to the aphelions
and nodes of the planets’ – meaning, the fixed stars are non-rotating with respect
to the aphelions and nodes, and therefore absolutely non-rotating.19 The second
reads:

And since these stars are liable to no sensible parallax from the
annual motion of the earth, they can have no force, because of their
immense distance, to produce any sensible effect in our system. Not
to mention that the fixed stars, everywhere promiscuously dispersed
in the heavens, by their contrary attractions destroy their mutual
actions, by Prop.lxx, Book 1.

On the first point, we must infer Newton never made the calculation Eq.(6); he
simply failed to see that the fall-off as the inverse square of the gravitational

18Eq.(2) is replaced by:
∇2φ + Λφ = −4πGρ.

Here Λ is the cosmological constant.
19Newton had, however, in Phenomenon I, II and IV, assumed the stars to be at rest. HIs

argument from induction for his theory of gravity was a complex one. See DiSalle [2006 Ch.2]
and , for the most comprehensive study to date, Harper [2012].

12



force with distance is cancelled by the quadratic increase in the number of stars.
But on the second the error is more subtle. Proposition LXX of Book 1 proved
that the gravitational force vanished everywhere in the interior of a uniform
shell of matter; Newton’s reasoning, then, would seem to be that the stars are
approximately uniformly distributed (‘promiscuously dispersed’), so their action
may be considered as that due to a concentric system of such shells with the sun
at their centre proceeding to infinity. Given all this, their influence is shown to
be zero.

We used Proposition LXX ourselves, in the case of an island universe, but in
that case the assumption that the solar system is at the centre of the universe
is clearly unwarranted. In the infinite case the assumption does not even make
sense. Since homogeneous and infinite, no point is at the centre. If one applies
the same construction anyway, one can build a system of spheres about any point
whatsoever, so points at completely arbitrary distances from the solar system.
Whereupon by the argument of Eq.(6) the acceleration can be anything and we
are back to Seelinger’s problem.

We are also back to Einstein’s problem. Such a universe cannot possibly be
static. Newton recognized rather better than did Einstein that here a balancing
act was impossible – that nothing short of divine providence could ensure a static
universe.20 That sounds like a reductio, but it is clear from correspondence with
Bishop Bentley that Newton welcomed the conclusion. Like Bentley, Newton
looked to science for arguments for the existence of God.

But our concern is with the definition of inertial frames, and on this point
Newton had more to say. The remark is to be found not in the Principia
itself but in an informal draft of Book 3, published posthumously in English
translation as The System of the World in 1728:

It may be alleged that the sun and planets are impelled by some
other force equally and in the direction of parallel lines, but by such
a force (by Cor.vi of the Laws of Motion) no change would happen
in the situation of the planets one to another, nor any sensible effect
follow: but our business is with the causes of sensible effects. Let us,
therefore, neglect every such force as imaginary and precarious, and
of no use in the phenomena of the heavens. . . (Cajori [1938 Sec.8].)

20Newton wrote:’That there should be a central particle, so accurately placed in the middle
as to be always equally attracted on all sides, and thereby continue without motion, seems to
me a supposition fully as hard as to make the sharpest needle stand upright on its point upon
a looking glass. For if the very mathematical centre of the central particle be not accurately
in the very mathematical centre of the attractive power of the whole mass, the particle will
not be attracted equally on all sides. And much harder is it to suppose all the particles in
an infinite space should be so accurately poised one among another, as to stand still in a
perfect equilibrium. For I reckon this as hard as to make, not one needle only, but an infinite
number of them (so many as there are particles in infinite space) stand accurately poised upon
their points. Yet I grant it possible, at least by a divine power; and if they were once to be
placed, I agree with you that they would continue in that posture without motion forever,
unless put into new motion by the same power.’ (Letter Jan. 17, Bentley [1838 p.208].)
Einstein’s solution (by means of a cosmological constant) required just such a balancing act;
small wonder he later called it ‘the greatest blunder of my life’ (on this point see Bianchi and
Roveli [2010]).

13



Corollary VI to the Laws is the statement:

COROLLARY VI

If bodies moved in any manner among themselves, are urged in the
direction of parallel lines by equal accelerative forces21, they will all
continue to move among themselves, after the same manner as if thy
had not been urged by those forces.

The proof was brief and there was no comment on it in the Scholium that
followed. It was used to prove Proposition III of section II (and Proposition
LVII and LXIV in later sections of Book 1); no other reference to it is to be
found in Principia. But Proposition IIIi was important to the argument for the
inverse square law of force for gravity in Book 3. The proposition is:

PROPOSITION III

Every body [A], that by a radius drawn to the centre of another body
[B], however moved, describes areas about that centre proportional
to the time, is urged by a force compounded of the centripetal force
tending to that other body, and of all the accelerative force by which
that other body is impelled.

That is, given the premise, the force by which body A is moved is compounded
of a centripetal force, and of whatever force is needed to make it accelerate at
the same rate as is B. But according to the Phenomena of Book 3 the moons of
jupiter do describe equal areas in equal times, so they must have those forces
acting on them, so in particular (apart from the force that makes them accelerate
at the same rate as jupiter) the moons experience a centripetal force towards
jupiter. This is the first part of Newton’s deduction from phenomena of the
force of gravity as centripetal and falling off as the inverse square.

But how is it that all the moons of jupiter are accelerated the same as jupiter
by the sun? The answer, of course, lies in Galileo’s equivalence principle – the
principle that in the absence of friction, all bodies are accelerated the same by a
uniform gravitational field, whatever their composition. In Newton’s terms that
meant the constancy of the ratio of gravitational mass µ to inertial mass m; and
it is to this that Newton immediately turned after establishing the force law of
gravity, at Proposition VI of Book 3.22 Newton argued for the universality of
this ratio at length, on purely empirical grounds, with reference to experiments
with pendulums that he had himself conducted.23

If, instead, the gravitational force is taken as given, the key to applying the
Laws to the jupiter system is not so much Corollary VI but the combination of
Corollary VI with Galileo’s equivalence principle, which may be stated as:

21Meaning, forces that produce equal accelerations (see Section 5).
22Although he had intimated it much earlier, at Definition VII of Book 1.
23An argument from induction, in Newton’s sense, meeting the requirements of his Rules

of Reasoning in Philosophy (in particular the third).

14



COROLLARY VI*

If bodies moved in any manner among themselves, are urged in the
direction of parallel lines by equal gravitational forces due to outside
bodies, they will all continue to move among themselves, after the
same manner as if they had not all been urged by that force.

In this by ‘equal gravitational force’ I mean equal force per unit gravitational
mass, denote −→g , so that acting on a body of inertial mass m and gravitational
mass µ the force is µ−→g . The resulting acceleration is

d2−→x
dt2

=
µ

m
−→g .

From the constancy of µ/m for any material body, whatever its constitution,
and given that −→g is uniform, the condition of Corollary VI is met. Since the
gravitational force of the sun is approximately uniform over the dimensions of
the jupiter system, we can conclude that the motions of jupiter and its moons
‘among themselves’ will be the same as if the centre of mass system were moving
inertially. By Corollary V, these are the same as if the centre of mass were at
rest, whereupon Newton’s laws apply.

It needs only a small rearrangement of this reasoning to conclude:

EQUIVALENCE PRINCIPLE

If bodies moved in any manner among themselves are described in
relation to an accelerating but non-rotating frame of reference, they
will all move in relation to that frame as if acted on by uniform
gravitational forces, producing the opposite acceleration.

With that we have a version of Einstein’s equivalence principle – and the germ
of an explanation for the universal proportionality of gravitational to inertial
mass. However, we are now remote not so much from Newtonian concepts,
as from his purpose in Book 3 of Principia – which was to deduce the inverse
square law from the observed motions.

Corollary VI in conjunction with Newton’s dynamical determinations of a
state of non-rotation and of equal time intervals24 solves the problem of inertial
frames for gravitational physics. The reason that NTM works in application
to the solar system, without ever giving an operational meaning to the notion
of inertial frame, is that none is needed: all that matters is that motions are
referred to a freely-falling non-rotating frame. That, to an excellent approxi-
mation, can be defined in terms of the dynamical behaviour of the solar system
– using Proposition XIV of Book 3 – regardless of the distribution of matter in
the rest of the universe. All that is needed is that the stars and galaxies are suf-
ficiently distant so as to give rise to gravitational fields approximately uniform

24At Proposition 1 Book 1, deriving Kepler’s second law (see Barbour [1989 pp.546-56] for
an illuminating discussion).

15



over the dimensions of the planetary orbits; whether or not they are changing
in time – and, indeed, whether or not the universe is infinite – is irrelevant.

But now notice Corollary VI* together with Corollary VI also solves the
problem of inertial frames for non-gravitational forces as well – so long as
they are described by Newton’s laws. The difference, on going over to non-
gravitational physics, is that there is no analog of the constancy of gravitational
to inertial mass; Corollary VI* cannot be used to justify the neglect of long-
range forces other than gravity. But precisely because charge-to-mass ratios of
every other force do vary, the existence of such forces would show up in the
relative motions of bodies, whether referred to an inertial frame or – and this is
the crucial point, by Corollary VI – to an accelerating but non-rotating frame.
So long as the latter (as a material system - say, a system of gyroscopes in
orbit around the earth) can be screened off from any non-gravitational forces
that might produce a torque, and so long as Newton’s laws apply, a dynamical
analysis will yield a criterion of nonrotation, and thereby define a nonrotating
frame. In principle such coordinates can be extended arbitrarily and used to
describe any bodies, anywhere in the universe.

What happens if the bodies comprising the freely-falling frame cannot be
screened off from nongravitational forces due to distant bodies producing a
torque? If there were such forces then there would be a difficulty. It may be,
in such a world, that nothing short of the centre of mass frame of the entire
universe would do – supposing it is a finite universe. In a world like that
it may well be that physics as we know it would not be possible. But that
world, thankfully, is not ours. The only long-range force other than gravity
is electromagnetism, which (except in extreme cases) can always be screened
off. And anyway, because charges of opposite sign tend to cancel, and those of
the same sign repel, we have good reason to believe there are no large remote
sources of this kind.

But if all of this is true, is there really any need of ‘real’ inertial frames, that
are anyway operationally unavailable? Why not make do with local, freely-
falling, non-rotating frames, as we do in GTR? If need be, the concept could
be idealized as the limiting case of such a frame. The idealization is safe, as in
practise such frames can be concretely realized (gyroscopes in orbit) to enormous
accuracy.25 It is also safe even if the univers is infinite: the further the mass
distribution, the more accurately its gravitational influence on the solar systtem
is approximated by a uniform gravitational field. Newton’s Principia, on our
reading, conforms to the maxim that theories be applicable whether or not the
universe is infinite. 26

That question carries with it another: what has any of this really to do
with gravity? Given that any rigid motion can be decomposed into a linear
acceleration and a rotation, all that really matters is that the frame of reference
be non-rotating. Free-fall, and the appeal to gravitational physics and screening

25The Gravity B Probe, recently used to test for dragging of inertial frames - hence as a
test of GTR - is an example.

26Note added Sep 2016. This is an optimistic reading. And insofar as it is right, there is no
guarantee of agreement with the non-relativistic limit of GTR: see Wallace [2016].

16



from all other forces, is simply the easiest way of materializing a non-rotating
frame. Corollary VI is more fundamental than the equivalence principle.

5. Kicking away the ladder

Of course our reasoning up to this point has been based on the laws (Axioms,
or Laws of Motion, Book 1), which suppose, by wide consent, that motions can
be referred to an inertial frame. Presented analytically, as from the time of
Lagrange, it is assumed there exist rectilinear inertial coordinates 〈−→x ,t〉 and
the equations written in terms of these. For N particles subject to forces F
that satisfy the third law (and that for simplicity are independent of relative
velocities), we have the system of equations:

mj
d2−→xj
dt2

=

N∑
k 6=j=1

F(−→xj −−→xk, t); j = 1, ...,N. (7)

The symmetry group of this system of equations (assuming the mj’s trans-
form as scalars) is the Galilean group (5). It would seem, then, that inertial
coordinates, in the usual Galilean sense, are needed to set up the theory ab ini-
tio. It may be that in practise we an refer all the motions to local freely-falling
frames, and the motions of the particles among themselves will all be the same
as if referred to an inertial frame, but this inertial frame has to be defined, at
least conceptually, to even write down the equations.

But the key concept to our reading of NTM as a relational theory, that of
‘motions among themselves’, plays a role here as elsewhere. It is most naturally
expressed in terms of difference equations for particle pairs, in terms of the
relative distance vectors

−→rjk = −→xj −−→xk. (8)

From the N equations (7), define the N(N − 1)/2 equations

d2−→rjk
dt2

=
1

mj

∑
l 6=j

F(−→rjl, t) −
1

mk

∑
l 6=k

F(−→rkl, t). (9)

Of these only N − 1 are linearly independent. From such a set, together with
the equation ∑

k

mk
d2−→xk
dt2

=
∑
k

∑
j 6=k

F(−→rjk, t) = 0 (10)

the N equations (7) can be derived, and vice versa. Eq.(10), the equation for
conservation of total momentum, follows from (7) and Newton’s third law (that
F(−→r jk, t) is antisymmetric in j and k. It is invariant under Galilean boosts

−→x k →−→xk + −→u t

17



(by Corollary V), as are the N(N − 1)/2 equations (9). But (8) and (9) are
invariant under the much wider class of symmetry transformations:

−→xk →−→xk +
−→
f (t) (11)

where
−→
f is any twice-differentiable vector-valued function of the time.27 The

imposition of Eq.(10) is what distinguishes ‘true’ inertial frames from our non-
rotating frames. If we dispense with (10), and allow that the system of equations
(8), (9) define the entire theory – say a new theory, different from NTM – the
symmetries of the equations will be those of Corollary VI, not just Corollary V.

There is, however, something odd about this way of putting it. For was
Corollary VI not already a consequence of the Laws, just as was Corollary V?
Why think Eq.(7) is the correct expression of Newton’s laws, rather than (8)
and (9)? The latter are clearly better suited to define the motions of a system
of bodies among themselves. If it was a check on the correct equations for
the expression of Newton’s laws that they exhibit the symmetries of the laws
(Corollary V), leading to (7), why isn’t it a check that they have the symmetries
of Corollary VI, namely (11), leading to (8) and (9) and not (7)?

The answer, presumably, will turn on the extent to which we can view the
Principia as at bottom a theory of the relative motions of particle pairs. As a
first step one would like to see if the Definitions and Laws can be rewritten in
a way that obviously translates into Eq.(8) and (9), rather than (7), and that
preserves intact the results and methods of proof used in the first two books of
Principia.

There is an obvious way to do this, namely, simply relativize them – sup-
pose that ’absolute space’ refers to any rigid nonrotating frame, in the same
way that in passing from Newtonian to neo-Newtonian space-time one takes
‘absolute space’ to refer to any inertial frame. No matter which frame is cho-
sen, the subsequent geometrical constructions (the main proof procedure used
in Principia) go through unchanged.

But that may seem like a cheat. Is it possible, rather, to interpret the laws
more directly as concerning the relative motions of particle pairs? I think that
it is – and to make clear how to do this, here are the laws in modified form,
keeping as close as possible to Newton’s language,28 the laws might be restated
as:

First Law* Every body continues in its state of relative rest, or uniform rel-
ative motion in a right line, with respect to another, unless compelled to
change that state by a difference in accelerative29 forces impressed upon
them.

27Eq.(11), together with the second of (5), were called the ‘Newtonian group’ by Ehlers
[1973].

28First law: ‘Every body continues in its state of rest, or of uniform motion in a right line,
unless it is compelled to change that state by forces impressed upon it’. Second law: ‘The
change of motion is proportional to the motive force impressed; and is made in the direction
of the right line in which that force is impressed’ (Cajori [1934, 12]).

29Note added 5 Sep 2016. Accelerative, and not motive, as incorrectly stated in [2013] (my
thanks to Tim Maudlin on this point).

18



Second Law* The change in the relative motion of two bodies is proportional
to the difference in accelerative forces impressed upon them, and is made
in the direction of the right line in which the difference in accelerative
forces is impressed.

Third Law To every action there is always opposed an equal reaction: or,
the mutual actions of two bodies upon each other are always equal, and
directed to contrary parts.

(The third law is unchanged.)
As stated, these laws clearly support (8) and (9) and of course the antisym-

metry of F . But now analagous modifications will have to be made in stating
some of their consequences, in particular Corollary IV. From the modified laws
above, it follows not that the centre of mass of all bodies acting upon each is in
uniform motion, ‘excluding external actions and impediments’, but that it is at
rest or in uniform motion relative to another (that is free from external actions
and impediments, or for which the relative external force vanishes). However.
the first part of Corollary IV stands unchanged.

But what does ‘uniform, straight line motion’ mean, if inertial motions (in
the usual sense of ‘inertial’) are undefined? Just this: relative straight-line
motions are those for which the relative distances (lengths of lines joining pairs
of bodies at each instant of time) and relative angles between lines joining
instantaneous bodies at various times, change in a characteristic way. What
way exactly? The way they change under the usual representation of force-free
bodies, when moving inertially.

In any case, whatever its relation to Principia, the system of equations
(8) and (9) together with the antisymmetry of F is of independent interest –
especially if, as seems to be promised, it yields exactly the same relative motions
as do (7) and (10). Can we kick away the ladder? Can we dispense with the first
equality of Eq.(10)? Formally there can be no obstacle: we are already assured
that any solution to the N equations (7) will yield N(N−1)/2 difference vectors
−→r jk satisfying (9). Conversely, for any solution to the N(N − 1)/2 equations
(9) satisfying (8) (of which only N −1 functions −→r jk are linearly independent),
choose

−→
f (t) so that its second derivative is

d2
−→
f

dt2
= −

1

M

∑
k

mk
d2−→xk
dt2

(12)

where M is the total mass, and apply the transformation (11). From the an-
tisymmetry of F Eq.(10) is then satisfied, and the coordinates as functions of
time then satisfy the N equations (7).

One can put it like this: the set of difference equations, Eq.(8) and (9), for
antisymmetric F , has a gauge freedom – an overall acceleration, constant in
space, as an arbitrary function of time.30 Up to this choice of gauge, solutions

30As a result Noether’s first theorem no longer applies, and the total momentum is not a
conserved quantity - as has already been observed.

19



to these equations are in 1:1 correspondence with solutions to (7).As Newton
said: ‘Let us, therefore, neglect every such force as imaginary and precarious,
and of no use in the phenomena of the heavens’.

6. Is it relationalism?

I have urged that a key concept of Principia is that of ‘motions among them-
selves’, and made this out in terms of difference equations. Is a theory based
on Eqs.(8), (9) and the symmetry of F a distinctively relational theory? It ap-
pears so. It is certainly a theory in which no meaningful notion of the velocity
or acceleration of a single body is available; there are only relative distances
and velocities of simultaneous bodies and their time derivatives, of two or more
bodies among themselves.

More than this is needed, however, if by ’relationalism’ we mean a genuine
eliminativism of reference to points of space or space-time. But just that is
what appears to be on the table: certainly the laws, as just modified, make
no mention of points of space or space-time; they speak only of quantities of
motion in terms of relations among bodies.

The devil, however, lies in the details. Let us examine these quantities more
carefully. As in Galilean-covariant theories, they include rates of change of the
norms of the −→r jk’s, denote rjk, the quantities:

drjk
dt

,
d2rjk
dt2

(13)

These require comparisons of spatial lengths at different times. Considering
three or more bodies, one also has rates of change of relative angles, as given by
(for unit rjk, rkl

cos−1(−→rjk ·−→rkl),
d

dt
cos−1(−→rjk ·−→rkl) (14)

Quantities of the form (13), (14) were always acceptable to relationalists.
The traditional difficulty has lain rather with rotations – say, two bodies

rotating about their common centre. If the motion is purely rotational, then
all quantities of the form (13), 14) are zero. Nevertheless the bodies have non-
zero relative velocities, and, referred to inertial frames, non-zero accelerations.
How do we describe their motion on our relationalist reading of Principia? The
answer is that the notion of relative velocity is independent of the principle
of inertia, so carries over unchanged, and the acceleration goes over to rela-
tive accelerations of the rotating bodies, with no need to refer it to an inertial
frame. The two bodies have a relative velocity because the direction of the line
connecting them changes in time, and they have a relative acceleration because
the direction of the relative velocity changes in time. The space-time structure
needed to make sense of rotations is that a spatial direction at one time can be
compared with a spatial direction at another time, with no need of the concept
of straight-line motion or departure from straight-line motion.

20



So much is obvious if we write the derivative as the limit of a difference
equation:

d

dt
−→rjk(t) = lim

∆t→0

1

∆t
[−→rjk(t + ∆t) −−→rjk(t)] . (15)

The mistake was to think this must be expressed as

d

dt
(−→xj −−→xk)(t) = lim

∆t→0

1

∆t
{[−→xj(t + ∆t) −−→xk(t + ∆t)] − [−→xj(t) −−→xk(t)]}

= lim
∆t→0

1

∆t
[−→xj(t + ∆t) −−→xj(t)] − lim

∆t→0

1

∆t
[(−→xk(t + ∆t) −−→xk(t)]

that is, as the difference between two velocities, each of which must be inde-
pendently defined. Only the relative velocity as given by Eq(15) is physically
meaningful, not velocities of which it is the difference.

Of course Machians go further, and reject quantities like (15) as well. The
comparison of spatial directions at different times is not needed, according to
Barbour [1999], no more than comparisons of time-like vectors in space-time.
The solution to a dynamical problem in Barbour’s terms is fully determined
given two relative configurations of particles at different times, without in-
formation on their relative orientation. Our form of relationalism is, from a
Machian point of view, a half-way house.31 Machians will press the epistemic
question: how are these angles between particle pairs at different times deter-
mined? Clearly not in the way angles between lines in space normally are –
the two lines cannot be simultaneously observed. The answer is, as always:
dynamically. The comparison (for a pure rotation) is in the first instance made
between the relative velocities of the two bodies at neighbouring times, divided
by the time interval – their relative acceleration. This quantity is given by a
centripetal force law in terms of masses and relative distances of bodies alone.
Since (for a pure rotation) their relative distance is constant in time, so is the
angular velocity and the norm of their relative velocity at each time. The rate
of change of relative velocity then determines the relative velocity (and the an-
gle between the lines joining the two bodies at two times) uniquely. All these
determinations (or rather their norms) are invariant under boosts to arbitrary
non-rotating frames, the transformations (11), as also, for time-independent
force-laws, under the transformations (5).

I suggest it was considerations like this that led Huygens to reconsider his
earlier view, in reluctant agreement with Newton, that cases of pure rotation
could only be understood as motions in relation to absolute space. At the end
of his life, he wrote instead:

31It is a halfway house from the point of view of Leibniz’s principles as well. Leibniz usually
spoke of temporal orderings rather than temporal intervals, but an ordering in time is still
a diachronic notion. If configurations of bodies at different times can be compared as to
order, why not compare them as to spatial direction as well? There is no reason – save
that, for Leibniz, all relations, temporal and spatial, are ultimately to be reduced to monadic
predications (they should be ‘internal’ relations). That is just what is provided (for temporal
orderings and intervals, if not spatial relations) by Machianism in Barbour’s sense.

21



Circular motion is relative motion along parallel lines, where the
direction is continually changed and the distance is kept constant
through a bond. Circular motion in one body is the relative motion
of the parts, while the distance remains constant owing to the bond.
(Huygens [1888-1950, Vol.21, p.507].)

Passages like these were quoted by Stein as evidence of the depth of Huygens’
thinking (Stein [1977]), but on this point there is hardly a consensus:

[T]o analyze rotation in terms of an objective, or absolute, notion
of velocity difference rather than objective, or absolute velocity is
to possess exactly the insight Newton lacked, but it is also to reject
the full-blown relational conception of motion, something that was
beyond the ken of Huygens’ philosophical dogmas. (Earman [1989
p.71].)32

Earman seems to equate Huygen’s insight with the shift from Newtonian to
neo-Newtonian space-time, but the insight is deeper: only a comparison of ve-
locities at the same time is needed, not at different times. We read Huygens
as dispensing with the principle of inertia. Earman thought no theory like this
was available in Newton’s day.

That claim, I have shown, is debatable. And whether or not a theory based
on (8) and (9) is a ‘full-blown’ relationalism, it is surely a theory in which:

True and simple motion of any one whole body can in no way be
conceived – what it is – and does not differ from rest of that body.
(Huygens [1888-1950, fragment 8, vol.16].)

7. Newton-Huygens space-time

The limiting process used in Sec.6 shows what is needed in terms of manifold
structures: a connection as a rule for comparing spatial directions at differ-
ent times – a rule for the parallel transport of directions in space (compatible
with the temporal and spatial metrics), not in space-time. Such a construction
(with the connection an ‘absolute’ structure) was given by Earman; he called it
‘Maxwell space-time’.33 Here I conclude with a construction that goes the other
way. Galilean space-time is the right manifold corresponding to neo-Newtonian
space-time, an affine-space; what does Maxwell space-time correspond to? It
should be a weaker structure – call it ‘Newton-Huygens space-time’ – but be
sufficient for the constructions used in Principia.

Its outline should already be clear. We are looking for a structure in which
affine notions and not just metrical ones are restricted to surfaces of constant

32Barbour [1989 p.675] is likewise critical of Huygens on this point.
33Earman [1989 §2.3, §4.7]. He still thought it inadequate to relationalism, however, as

flouting his relational requirement R1, that space-time ‘cannot have structures that support
absolute quantities of motion’. (But whether rotation is absolute or relative is part of what
is here in contention.)

22



time. Define Newton-Huygens space-time as the structure 〈X ,V3,V1, +3, +1〉,
where

• V3and V1 are real vector spaces of dimensions 3 and 1 respectively, with
Euclidean inner products 〈., .〉3, 〈., .〉1, respectively.

• +3 is a free (but not transitive) action of V3 on X , i.e. for any a ∈X ,
−→
b

∈ V3, a +3
−→
b ∈X

• +1 is a free and transitive action of V1 on the cosets of X under the action
of V3.

The action of V3 on X defines an equivalence relation on X , ‘simultaneous with’;
the cosets of X under V3 are time-slices. By construction, +3 acts transitively
on any time-slice. Hence, since +3 is free, for any simultaneous a,b ∈X , there
is a unique −→c ∈ V3 such that a +3 −→c = b, denoted b − a ∈ V3. Now consider
two pairs of simultaneous points in X . If a is simultaneous with b, and a′ is
simultaneous with b′ (possibly at a different time), the quantity:

cos θ =
〈(a− b), (a′ − b′)〉3
|a− b|3|a′ − b′|3

is well-defined. θ is the angle between the directions in space a− b and a′ − b′.
But there is no notion of straight line connecting non-simultaneous points; the
principle of inertia does not apply.

There remain important questions, above all, moving over to a manifold for-
mulation: What is the relation between a theory of gravity and other forces for-
mulated in Maxwell space-time and one based on Newton-Cartan space- time?34

In the latter there is a notion of parallelism for time-like as well as space-like
vectors. That notion, we must conclude, is dispensable, to be derived, if at all,
by fixing of gauge.

That takes us, again, to field theory, which is beyond our remit here. But it
raises a conceptual worry. It might be thought that insofar as NCT is approx-
imately empirically adequate, as go the motions of the planets and terrestrial
gravitating bodies, there is de facto an empirical standard of parallelism for
time-like vectors. A theory based on Maxwell space-time will simply be unable
to account for this standard.

That argument, however, is too quick. We have already demonstrated the
empirical equivalence of a theory based on Newton-Huygens space-time to one
based on neo-Newtonian space-time, although the latter too incorporates a stan-
dard of parallelism for time-like vectors. That standard, from the perspective of
Newton-Huygens space-time, is not an empirical given: it is unneeded surplus
structure. We have already shown how to obtain it in the case of a finite uni-
verse by a choice of gauge, Eq.(12). Something similar follows from the point
of view of Barbour-Bertotti space-time, in which Eq(7) can be derived by a
certain parameterization of the allowed curves in relative configuration space.

34This question was recently addressed (and partially answered) by Weatherall [2015].

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And of course, according to that theory, the standard of parallelism of space-
like vectors, with which Newton-Huygens theory is equipped, is likewise surplus
structure, unnecessary to the empirical adequacy of the theory (given that the
universe has zero total angular momentum and energy).

So much for the actual world. What of possible worlds, and distinctions
among them drawn in NCT, invisible to ours? Take possible worlds each with
only a single structureless particle. Depending on the connection, there will be
infinitely many distinct trajectories, infinitely many distinct worlds of this kind.
But in Newton-Huygens terms, as in Barbour-Bertotti theory, there is only one
such world – a trivial one in which there are no meaningful predications of the
motion of the particle at all. Only for worlds with two or more particles can
distinctions among motions be drawn. From the point of view of the latter
theories, the fault lies with introducing a nontrivial connection – curvature –
without any source, unrelated to the matter distribution. At a deeper level, it
is with introducing machinery – a standard of parallelism for time-like vectors,
defined even for a single particle – that from the point of view of a relationalist
conception of particle motions is unintelligible.

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