Isaac Newton was born in 1642, the year Galileo Galilei died, and he died in 1727. The range of Newton's activity was far reaching. Not only was he an eminent scientist, but he was also interested in biblical criticism and, later in life, he became a functionary in the British government as head of the Royal Mint. The last 22 years of his life he was the president of the Royal Society, the epitome of establishment science.

His scientific achievements are indeed impressive. By the time he was 24 years old, he had made three discoveries, any one of which would have garnered him a Nobel Prize, had such an institution been in effect at the time. The year before, he had discovered the binomial theorem. Newton's own recollection of these days, written in 1718, is that: In the beginning of the year 1665 I found the method of approximating series and the rule for reducing any dignity [power] of any binomial into such a series. [The Binomial Theorem] The same year in May I found the method of tangents of Gregory and Slusius, and in November had the direct method of fluxions [the differential calculus] and in the next year [1666] in January had the theory of colors [Newton's Theory of Colors] and in May following I had entrance into the inverse method of fluxions [the integral calculus]. And the same year I began to think of gravity extending to the orb of the moon . . . All this was in the two plague years of 1665 and 1666, for in those days I was in the prime of my age for invention and minded Mathematics and Philosophy more than at any time since. (Hall, 1980, pp. llf.)

Later biographers of Newton were to refer to the year 1666, when Newton was 24,
as the __Annus__ __Mirabilis__, "the marvelous year." In that year, the
plague broke out in London forcing Newton to retire to Lincolnshire where he
spent his time in reflection and experimentation. Newton's theme developed his
theory that light had a "mixed nature," that is, that white light was a
combination of distinct colored rays which could be decomposed by a prism. The
simple and elegant experiments upon which this conclusion was based were
published in the Philosophical Transactions in 1672. Newton's views were the
source of much dispute among his contemporaries, and Newton, who was a very
private man, was led to be very cautious about what he would commit himself to
publicly (For a discussion of 17th century optics and the controversy over
Newton's views, see Sabra, 1967.).

The second major achievement of Newton's during this period was the invention of the calculus. From antiquity, many mathematicians had discovered various rules for calculating the areas and volumes of special objects, such as the sphere and the cone. Archimedes, for example, discussed several methods for estimating the volumes of various regularly shaped objects and also for estimating the areas of conic sections. These rules for calculating areas and volumes form the basis of a rudimentary integral calculus. Newton's contribution was to codify these methods into a single unified comprehensive method which could be applied generally, and not merely to special cases. He also worked on general methods for calculating the rates at which functions change. Again, prior to Newton's time, special rules had been developed for calculating rates of change for special functions. Newton's contribution, again, was to provide a codified systematization of these rules and to formulate a general procedure whereby rates of change could be calculated, in principle, for any arbitrary functional formula. Although Newton did this in 1666, when he was 24, he did not publish the results or suggest to anyone that he had accomplished these tasks until much later. This hesitancy on his part led to a bitter priority dispute between Gottfired Wilhelm Leibniz and himself. Leibniz, of whom we will say more in the next chapter, developed and published his own version of the calculus before Newton. Leibniz had access to some of Newton's correspondence in 1676, and later, Newton tried to establish that Leibniz had stolen the basic idea of the calculus from him. This led to a bitter priority dispute which raged from 1710 to Leibniz's death in 1717, and which spilled over into a dispute about fundamental philosophical principles and about the nature of space and time. The general consensus today is that the calculus was developed independently by both Leibniz and Newton (a recent book by A.R. Hall, 1980, is a detailed study of the dispute).

Newton's third basic discovery was his conjecture that the moon was held in its
orbit by __gravity__, i.e., by the same force that pulled apples to the
earth was also pulling the moon to the earth. This was a key idea in Newton's
subsequent development of the principles of classical mechanics. Among other
things, it marked the end of the Aristotelian model which had divided the
universe into two parts, the heavens and the sub-lunar realm, each governed by
its own set of laws. Newton's insight eventually led him to formulate a theory
of mechanics which postulated one set of laws for all mechanical phenomena
anywhere in the universe.

Again, Newton was hesitant to publish his results and, in fact, he did not
mention them to anyone for another 20 years. Various explanations for Newton's
reticence have been offered. The standard story is that Newton was worried
about the fact that the predictions of his theory did not agree closely enough
with the actual measured orbital velocities and positions of the moon. The
mistake turned out to be an error in the measurement of the size of the earth
rather that an error in Newton's mechanics. Even so, the treatment of the
motion of the moon remained incomplete even in the published version of
Newton's theory, __The__ __Mathematical__ __Principles__ __of__
__Natural__ __Philosophy__ [__Principia__].

Newton was moved to publish his views in response to a question from the
astronomer Edmond Halley, who was the clerk of the newly formed Royal Society
of London. Halley served as a go-between communicating results from one
scientist or philosopher to another (in those days communications were
difficult and there were few journals). I.B. Cohen gives the following account
of what happened: The history of the __Principia__ begins with a definite
event: a trip from London to Cambridge to see Newton, made by Edmond Halley,
one of the secretaries of the Royal Society, famous today primarily for his
contributions to astronomy and for the comet named after him, but then also
considered an able geometer. The date is 1684, presumably August. Newton, aged
41, is Lucasian Professor of Mathematics at Cambridge and a fellow of Trinity
College, known at large for his published discoveries concerning light and
color, but admired by the cognoscenti for his work in mathematics (not as yet
published, but to some degree circulated in manuscript) and his grasp of the
fundamentals of dynamics. During the visit, Halley asks Newton what path the
planets would describe if they were continually attracted by the sun with a
force varying inversely as the square of the distance. Newton replies that the
path would be an ellipse. This is not a mere guess on Newton's part, but a
result he has obtained by mathematics, a proved statement. Newton--to continue
with the story--cannot lay his hands on the calculation,' but promises to send it
to Halley in London. He
reworks the material and composes a short tract which Edward Paget takes to
London to give Halley. On reading it. Halley gets so excited that he returns to
Cambridge to see Newton once again. One effect of his visit is that Newton
promises to make public his work. He accordingly sends a tract (presumably the
one previously shown to Halley) to the Royal Society. Encouraged by Halley, and
by the warm commendations of the Royal Society, Newton eventually completes a
manuscript for publication. Such are the beginnings of the __Principia__.
(Cohen, 197~, 47)

The __Principia__ is an axiomatic treatment of the principles of mechanics.
In it, Newton starts out by laying down certain definitions and assuming
certain postulates, the "three laws of motion." From these laws and the
definitions, Newton deduced a number of theorems concerning the motions of
various bodies and the forces under which they were acting. Despite the fact
that Newton had invented the calculus, he did not use the calculus in writing
the __Principia,__ but rather wrote in the geometrical style which was more
familiar to his contemporaries. The result is that all the proofs in the
__Principia__ are done in what appears to us today to be a tedious and
obscure manner. Even so, the __Principia__ is a difficult book, and was only
slowly accepted by other natural scientists, especially those who worked
outside England. Newton was an extremely complex person. He never married, nor is
there
any indication that he had much to do with women at all. He was involved in
several controversies about scientific priority during his lifetime. He was
involved in a bitter dispute which the physicist Robert Hooke on the question
of who had first discovered that the gravitational force was an inverse square
law. Finally, there were the bitter disputes with Leibniz. It started with a
dispute over priority with respect to the invention of the calculus. Leibniz
then vigorously challenged Newton's conception of space and time and, in
particular, his analysis of gravity, which, for Leibniz, amounted to the
reintroduction into physics of __occult__ qualities. This latter dispute
produced a correspondence between Leibniz and a spokesman for Newton, Samuel
Clarke, which continued from 1713 to Leibniz's death in 1717. After Leibniz's
death, it was published and is known today as the Leibniz-Clarke correspondence
(see Alexander, 1J56).

Newton, as is characteristic of many great men, was both arrogant and modest. He once compared himself to a small boy playing on the shore by an ocean of truths only some of which he had explored. In a letter to Hooke, Newton said, "If I have seen farther [than my fellow man] it is because I have stood on the shoulders of Giants." Hooke was supposed to get the impression that he was perhaps one of the giants on whose shoulders Newton stood.

In 1703, Newton became the president of the Royal Society, over which he reigned for the next 24 years until his death in 1727 (see Westfall, 1980, for the most definitive biography of Newton).

II. __The Newtonian System__

Before discussing the Newtonian concepts of space and time, we should try to
get clear about the Newtonian system itself. What Newton developed in the
Principia was to serve as the foundation of classical mechanics until the 20th
century, but although widely acclaimed when it was first published, it did not
command immediate universal consent. When Newton began to write the
__Principia__ in 1684, 61 years had passed since Galilei had called for a
mathematical representation of natural phenomena (Drake, 1956, 237ff~.
Describing natural phenomena in mathematical terms is one task; explaining
those phenomena by means of mathematical principles is another. Galilei had
himself produced, of course, several mathematical laws or principles, among
them, the law of falling bodies: s = 1 /2 gt2 which related the distance (s)
that an object falls in a time (t), where (g) is the acceleration due to
gravity. The problem with Galilei's approach is that it is too particular. The
law of falling bodies which Galilei formulated is restricted in several ways:

(l) it deals only with freely falling bodies;

(2) it deals with the motion under the influence of one special force, gravity;

(3) it assumes that the force of gravity is a constant.

None of this is meant to take anything away from the magnitude of Galilei's achievement which was great indeed, but only to illustrate Galilei's approach, which was to formulate particular principles for particular phenomena. If a full fledged mathematical philosophy was to develop, however, it was clear to some of Galilei's contemporaries that a more general approach was required (see the discussion in chapter 6, pp.6-43f). Descartes, in particular, argued that what was needed was a set of general mathematical principles which applied to all motions under any circumstances. Inspired by the deductive ideal of Euclidean geometry, Descartes hoped to produce a small set of mathematical principles of natural phenomena (corresponding to Euclid's axioms) from which descriptions of all natural phenomena could be deduced (corresponding to Euclid's theorems). He proceeded to do so and, in 1644, he produced a list of 3 laws of nature and 7 rules of impact which formed the basis of what became known as Cartesian physics. The laws are:

### FIRST LAW: Each thing, in so far as in it lies, always perseveres in the same state, and when once moved, always continues to move.

- SECOND LAW: Every motion in itself is rectilinear, and therefore things which
are moved circularly always tend to recede from the center of the circle which
they describe.
- THIRD LAW: If a moved body collides with another, then if it has less force to continue in a straight line than the other body has to resist it, it will be deflected in the opposite direction and, retaining its own motion, will lose only the direction of its motion. If it has a greater force, then it will move the other body along with itself and will give as much of its motion to that other bodies as it loses.

- RULE (l) If two bodies B and C are completely equal and are moved with equal
velocity, B from right to left and C from left to right, then when they
collide, they are reflected and afterward continue to be moved, B toward the
right and C toward the left,
without losing any part of their velocities.
- RULE (2) If B is slightly larger than C, and the other conditions above still
hold, then only C is reflected and both bodies are moved toward the left with
the same velocity.
- RULE (3) If they are equal in size, but B is moving slightly faster than C,
then not only do they both continue to be moved toward the left but also B
transmits to C part of its velocity by which it exceeds C. Thus, if B
originally possessed six degrees of velocity and C only four, then after the
collision they both tend toward the left with five degrees of velocity.
- RULE (4) If C is completely at rest and is slightly larger than B, then no
matter how fast B is moved toward C, it will never move C but will be repelled
by C in the opposite direction. For a body at rest gives more resistance to a
larger velocity than to a smaller one in proportion to the excess of the one
velocity over the other. Therefore there is always a greater force in C to
resist than in B to impel.
- RULE (5) If C is at rest and is smaller than B, then no matter how slowly B is
moved toward C, it will move C along with itself by transferring part of its
motion to C so that they are both moved with equal velocity. If B is twice as
large as C, it transfers a third of its motion to C because a third part of the
motion moves the body C as fast as the two remaining parts move the body B
which is twice as large. And thus, after B has collided with C, B is moved one
third slower than it was before, that is, it requires the same time to be moved
through a space of three feet. In the same way if B were
three times larger than C, it would transfer a fourth part of its motion to C,
etc.
- RULE (6) If C is at rest and is exactly equal to B, which is moved toward C,
then C is partially impelled by B and partially repels B in the opposite
direction. Thus, if B moves toward C with four degrees of velocity, it
transfers one degree to C and is reflected in the opposite direction with the
remaining three degrees.
- RULE (7) Let B and C be moved in the same direction with C moving more slowly and B following C with a greater velocity so that they collide. Further, let C be greater than B, but the excess of velocity in B is greater than the excess of magnitude in C. Then B will transfer as much of its motion to C so that they are both moved afterward with equal velocity and in the same direction. On the other hand, if the excess of velocity in B is less than the excess of magnitude in C, then B is reflected in the opposite direction and retains all of its motion. These excesses are computed as follows. If C is twice as large as B but B is not moved twice as fast as C, then B does not impel C but is reflected in the opposite direction. But if B is moved more than twice as fast as C, then B impels C. For example, if C has only two degrees of velocity and B has five, then C acquires two degrees from B which, when transferred into C, become only one degree since C is twice as large as B. And thus the two bodies B and C are each moved afterward with three degrees of velocity. And other cases must be evaluated in the same way.

When integrated with Descartes' dualistic philosophy, Cartesianism seemed to provide the promise of a unified philosophical approach to the world which could replace the weakened and discredited Aristotelian philosophy. Newton's own ideas about physics were developed, in part, in reaction to the principles of Cartesian physics. As Newton and other contemporaries saw, there are a number of difficulties with interpreting and applying Descartes' physical laws and rules.

(l) First, there is a problem about the relation between the laws and rules. The rules are supposed to be derivable from the laws, but how this is to be accomplished is by no means clear. In addition, the system fails to have the required generality. The rules, which do all the actual work, apply only to special cases. The laws, of course, are general, but do not seem to be readily applicable.

(2) Many of the concepts that appear in both the laws and the rules are unclear. Descartes never provides unambiguous and clear cut rules for interpreting such terms as "force," "quantity of motion," and "resistance." Without such guidelines, the applicability of the rules to particular cases if not clear.

(3) Despite the Cartesian goal of a mathematical physics, Descartes' principles are not very mathematical.

(4) Where the rules to admit of clear mathematical interpretation, they are often wrong. Consider RULE (4), for example. If it were correct, then a speeding bullet would not be able to move a heavier, stationary object. But moving bullets are quite efficient at doing just that, so RULE (4) cannot be right as it stands.

(5) A deeper problem is that there is no systematic role for the concept of relative motion in Cartesian physics. This can be seen quite clearly by reflecting on the phrase "velocity in B," e.g., as it appears in RULE (7). A careful reading of the rules makes it clear that, for Descartes, velocity is an absolute property of a body. This contradicts the Galilean principle of relativity (of which more below; see p.7--45f). Newton saw the failure of Cartesian physics to do justice to the relational character of motion as a fundamental error (For discussion of these criticisms and others, see Blackwell, 1966).

Faced with these difficulties, and given the powerful mathematical tools which he had developed, Newton set out to produce a better set of mathematical principles. Having produced it, it was widely acclaimed by some (e.g., Halley, and, in general, the English scientists) and rejected by others (e.g., Leibniz, and, in general, the continental physicists, although there were notable exceptions). The result was that the Newtonian system struggled with the Cartesian system into the 1700's. Well before 1750, however, almost all (respectable) physicists were Newtonians.

Essentially, Newton's theory is a theory of particle mechanics, that is, a theory which describes and predicts the behavior of particles and particle systems as they evolve through time. The immense success of the Newtonian system in explaining not only the motion of the planets, but also the motion of pendula, the oscillation of springs, and a whole host of other physical phenomena led certain investigators to think that the principles of Newtonian mechanics held the ultimate key to the understanding of all physical phenomena. In the 18th century this contributed to the development of a movement which held, in effect: every natural system is a Newtonian particle system. The ideal was to reduce all branches of physical knowledge to Newtonian mechanics. The ideal remained only an ideal as one field after another in the l9th century failed to be reduced to Newtonian mechanics, until, in the 20th century, the development of the theories of relativity and quantum mechanics superseded Newtonian mechanics itself.

Newton's theory rests on three laws of motion plus a law of gravitational
attraction. The three laws of motion are set forth at the beginning of Book 1
of the __Principia__, "__The__ __Motion__ __of__ __Bodies__."
The law of gravity appears in Book III. "The System of the World." The laws
are:

- LAW I:
__The____Principle____of____Inertia__Every body continues in its state of rest, or of uniform motion [= constant velocity] in a right [= straight] line, unless it is compelled to change that state by forces impressed upon it.
Compare this principle to the first two laws of the Cartesian system. The
content of the two formulations (Newton's and Descartes') is the same but
Newton's is more readily applicable (see the next law). On the basis of this
similarity, Descartes is credited with having anticipated the principle of
inertia. - LAW II.
__The____Definition____of____Force__The change of motion is proportional to the motive force impressed; and is made in the direction of the light [straight] line in which the force is impressed.
This principle can be given a precise mathematical formulation: - LAW III:
__The____Action--Reaction____Principle__To every action there is always opposed an equal reaction; or the mutual actions of two bodies on each other are always equal, and directed to contrary parts.

F mv where "F" is the impressed force, and "mv" is the motion of the body.
Another key difference between Descartes and Newton appears here. For Newton,
the key kinematic (i.e., dealing with motion) concept is __acceleration__ (=
rate of change of velocity v), whereas for Descartes, as a perusal of his rule
shows, the key kinematic concept is __velocity__.

In Book III, Proposition VII, Theorem VII concerning the gravitational attraction that all bodies were postulated to have for each other, Newton writes That there is a power of gravity pertaining to all bodies, proportional to the several quantities of matter which they contain.

COROLLARY II: The force of gravity towards the several equal particles of any body varies inversely as the square of the distance of places from the particles.

The statement and corollary can be formulated as follows:

where "F" is the force of gravitational attraction between two particles or bodies A and B, "G" is a universal constant, "M" is the mass of body A, "M" is the mass of body B, and "r^2" is the square of the distance separating the particles or the center of masses of the two bodies.

The three laws form the foundation of Newton's theory of particle mechanics. From these first principles, the motion and behaviors of a large number of physical systems can be predicated and described with great accuracy.

As we said before, the great power and enormous success of the Newtonian system suggested to many people that the entire universe was a huge mechanism each part of which could be explained completely and with great precision by these mechanical principles. Although the approach was enormously successful, it was never completely successful. From the very beginning, there were questions about some of the fundamental concepts upon which the Newtonian system was based. Newton's view that space and time were an absolute background for the mechanical motions described and explained by the theory was challenged by Leibniz and George Berkeley.

Newton's conception of gravity was also challenged. Leibniz saw the Newtonian
concept of gravity, which involved forces acting on bodies which were separated
from one another (action at a distance), as an appeal to "occult qualities,"
that is, to mysterious "I-know-not-whats" which were being invoked by Newton to
account for physical phenomena. As such, Leibniz saw it as a reversion to the
discredited approach of the medieval philosophers. The Cartesian program, which
Leibniz followed, had as its aim the explanation of the motion of bodies solely
in terms of contact forces, on the plausible ground that the influence of one
body on another was more "intelligible" if the bodies were in contact but more
or less mysterious if they were not. In his public response to these
criticisms, Newton distinguished between the __mathematical__
__description__ of gravitational phenomena (the Law of Gravity) from an
__interpretation__ or __explanation__ of what constituted the
essence of gravity. In the General Scholium to Book III, added to the second
edition, Newton writes:

Hitherto we have explained the phenomena of the heavens and
of
the sea by the power of gravity, but have not yet assigned the cause of this
power. This is certain, that it must proceed from a cause that penetrates to
the very centers of the sun and planets, without suffering the least dimunition
of its force; that operates not according to the quantity of the surfaces of
the particles upon which it acts (as mechanical causes used to do [a reference
to Descartes and his view that the important quantity was the size and not the
mass of a body?]), but according to the solid matter which they contain, and
propagates its virtue on all sides to immense distances, decreasing always as
the inverse square of the distances. . . hitherto I have not been able to
discover the cause of those properties of gravity from phenomena, and I frame
no hypotheses [__hypotheses__ __non__ __fingo__]; for whatever is not
deduced from the phenomena is to be called an hypothesis; and hypotheses,
whether of occult qualities or mechanical, have no place in experimental
philosophy. In this philosophy particular propositions are inferred from the
phenomena, and afterwards rendered general by induction. Thus it was the
impenetrability, the mobility, and the impulsive force of bodies, and the laws
of motion and of gravitation, were discovered. And to us it was enough that
gravity does really exist [as evidenced by its effects], and act according to
the laws which we have explained, and abundantly serves to account for all the
motions of the celestial bodies, and of our sea [compare with Galilei].
(Newton, 1962, pp. 546f.)

This is Newton's public face. Privately, in a letter to Richard Bentley, written February 25, 1692-1693, but unpublished until much later, Newton wrote: It is inconceivable that inanimate brute matter, should, without the mediation of something else, which is not material, operate upon and affect other matter without mutual contact, as it must be, if gravitation, in the sense of epicurus be essential and inherent in it. And this is one reason why I desired you would not ascribe innate gravity to me. That gravity should be innate, inherent, and essential to matter, so that one body may act upon another at a distance through a vacuum, without further mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity, that I believe no man, who has in philosophical matters a competent faculty of thinking, could ever fall into it. Gravity must be caused by an agent acting constantly according to certain laws; but whether this agent be material or immaterial, I left to the consideration of my readers. (Newton, 1953, p. 54)

What are we to make of this? The point is that Newton is trying to carefully
distinguish between the mathematical expression of a law and its explanation.
What he has produced, and all that a working physicist needs, is a mathematical
formulation of the effect of gravity. This is what he uses to describe and
predict the motion of the heavenly bodies. He is with Galilei on this point in
urging that the first concern of the physicist should be the production of
mathematical laws of nature. What hypothetical understanding of these laws we
have is of secondary concern. Nevertheless, Newton is unhappy with this state
of affairs. He is working in the "contact force" traditional of 17th century
physics and is willing to admit, in private at least, that there is something
more that needs to be said about gravity. What needs to be said, however, is
not, in Newton's view, part of the mathematical principles of natural
philosophy. In his heart of hearts, Newton hoped that his readers would be led
to the conclusion that the causal agent producing gravity was, ultimately,
God. For Newton, gravity and the principles of the __Principia__ in general
were as much evidence of God's working in the universe as the Scriptures.

III. __Newtonian Space and Time__

We turn now to a consideration of the
general characteristics of Newton's view
of space and time. Although Newton does define some of the basic terms that he
employs, for example, "mass," "force," and "acceleration," he does not define
"time," "space," "place," and "motion" since he takes these "as being well
known to all." In the Scholium to the definitions preceding Book I, however,
Newton does offer some remarks about time, space, place and motion. In
particular, he is concerned to distinguish between __absolute__,
__true__, or __mathematical__ time (space~, on the one hand, and
__relative__, __apparent__ or __common__ time (space), on the other.
Relative space, relative time, and relative motions are spaces, times and
motions which are measurable by us. . All actual measurement, for Newton,
involves comparing one thing with another. Thus, when one uses a measuring tape
to determine that a room, say, is 12 feet wide, 20 feet long and 8 feet high,
what one has determined is a relative space, since the measured space of the
room is what it is in virtue of its relation to the tape as standard. The
measurement of time involves comparing the duration of a process or activity to
the duration of an interval ticked off by our clocks. Similarly, objects are
determined to be in motion only if they move with respect to something or
other.

However, in addition to these relative measures, Newton believed and argued for the existence of Absolute space, Absolute time, and Absolute motions. These absolute quantities are insensible, that is, they cannot be directly detected by human observers, yet Newton felt that they had to exist. The plausibility and necessity of these absolutes was attacked vehemently by relationalist critics of Newton such as Leibniz and Berkeley (see the discussion in the next chapter). We discuss Newton's reasons for holding that such absolutes did exist in the course of fitting his views on space and time into our categorical framework.

1. __Absolute__ __space__ __and__ __absolute__ __time__
__are__ __qualitatively__ __homogeneous__.

In the Scholium, Newton says, "Absolute space, in its own nature, without relation to anything external, remains always similar and immovable." That absolute space "remains always similar" is just to say that its parts are qualitatively homogeneous.

Even so, in an unpublished manuscript, Newton puts the case more strongly: "The parts of duration and space are only understood to be the same as they really are because of their mutual order and position; nor do they have any hint of individuality apart from the order and position which consequently cannot be altered (Hall and Hall, 1962, p. 136)." This contrasts with the Aristotelian view, at least with respect to space, since Aristotle, recall, held that space was qualitatively heterogeneous.

2. __Absolute__ __space__ __(time)__ __is__ __infinite__
__(Eternal).__

Newton's arguments for the infinite extension of space and time (duration) are both conceptual and theological in nature. Newton says: Space extends indefinitely to all directions. For we cannot imagine any limit anywhere without at the same time imagining that there is a space beyond it. And hence all straight lines, paraboloids, hyperboloids, and all cones and cylinders and other figures of the same kind continue to infinity and are bounded nowhere, even though they are crossed here and there by lines and surfaces of all kinds extending transversely, and with them form segments of figures in all directions. (Hall and Hall, 1962, pp. 133-134)

Similarly, in the same manuscript, Newton argues that we cannot think "that
there is no duration" (Hall and Hall, 1962, p. 137).
The first quotation indicates that Newton is thinking of physical space as a
realization of a three-dimensional Euclidean space. In addition, Newton seems
to have some theological reason for thinking that space is infinite in extent
and eternal in duration (this latter claim, that space is eternal in duration
entails, of course, that duration is itself eternal). According to this same
manuscript, Newton held the view that space and time were properties of being
__qua__ being, by which he meant that no being could exist which was not
related to space and time in some way. Since Newton believed that God was an
eternal and infinite being, it followed that space must be infinite in extent
and time (or duration) infinite in extent as well (Hall and Hall, 1962, p.
136).

These unpublished remarks of Newton are somewhat obscure and confusing.
However, even the published __Principia__ contains theological
considerations with respect to the nature of space and time. For example, in
the General Scholium to Book III, Newton writes: . . From his true dominion it
follows that the true God is a living, intelligent, and powerful Being; and
from his other perfections, that he is supreme, or most perfect. He is eternal
and infinite, omnipotent and omniscient; that is, his duration reaches from
eternity to eternity; his presence from infinity to infinity; he governs all
things, and knows all things that are or can be done. He is not eternity and
infinity, but eternal and infinite; He is not duration or space, but He endures
and is present. He endures forever, and is everywhere present; and, by existing
always and everywhere, he constitutes duration and space. Since every particle
of space is __always__, and every indivisible moment of duration is
__everywhere__, certainly the Maker and Lord of all things can not be
__never__ and __no__ __where.__ (Newton, 1962, p. 545)

3. __Space__ __and__ __Time__ __are__ __Continuous__.

The continuity of time follows from Newton's assertion in the Scholium that "Absolute, true, and mathematical time, of itself, and of its own nature, flows equally without relation to any thing external . . .." That Newton takes absolute time to "flow equably" is a good indication that he thought that the flow of time was continuous and not discrete. We must, of course, remember that, in Newton's time, to say that an interval was continuous was to say that it was dense but not point-like.

If we are correct in thinking that Newton implicitly took physical space to be a realization of 3-dimensional Euclidean space, then the continuity (as opposed to discreteness) of space follows. This is so because, as we noted earlier in our discussion of Zeno's paradoxes, certain theorems which are true of Euclidean spaces (and, hence, by hypothesis, of physical space), e.g., the Pythagorean theorem, fail to hold for discrete spaces.

4a. __Absolute__ __space__ __is__ __isotropic__.

The isotropy of absolute space follows from Newton's treatment of absolute space as an infinite, qualitatively homogeneous continuum. Given that all the spatial points are qualitatively identical, there is no way for certain directions, as such, to be preferred. This contrasts, recall, with the Aristotelian position which, because space was finite and because of Aristotle's doctrine of natural places, treated space as anisotropic (i.e., with intrinsic preferred directions).

4b. __Absolute__ __time__ __is__ __anisotropic__.

Newton does not have anything to say about this particular property of time, taking it, one might suppose, as one of those properties of time which is "well known to all." It might be well to remind ourselveg at this point that the anisotropy of time is not a feature which is assumed by or derivable from the Newtonian laws of mechanics. The laws of mechanics for particles or particle systems ~LAWS I-III] are symmetric with respect to time. This means, in effect, that the laws of mechanics are indifferent to whether mechanical processes proceed in one temporal direction or the other; were time to somehow "run backwards" (from the "future" to the "past"), the laws of mechanics would not be violated. Hence, no mechanical experiment could detect such a "time reversal." From the point of view of mechanics, then, the "directionality" of time is somewhat of a mystery. Some people have tried to account for the unidirectionality of time by appealing to the laws of thermodynamics. (For such an attempt, see Reichenbach 1956. For other discussions, see Grunbaum, 1973 and van Fraassen, 1970. There is an extensive bibliography on the topic in Gale, 1967. For reservations about the thermodynamic approach, see Melhberg, 1980.)

5. __Absolute__ __space__ __and__ __absolute__ __time__
__are__ __object__ __independent__.

What this means is that even if there were no objects, absolute space would
still exist as a receptacle or container within which objects __could__ be
placed. Similarly, absolute time exists and would exist even if there were
nothing changing nor any object enduring. Thus, Newton says: ...Although we can
possibly imagine that there is nothing in space, yet we cannot think that space
does not exist, just as we cannot think that there is no duration, even though
it would be possible to suppose that nothing whatever endures. (Hall and Hall,
1962, p. 137)

These sentiments are expressed by Newton in the Scholium to the definitions in
the __Principia__ itself. There Newton says, "Absolute, true, and
mathematical time, __of__ __itself__, and from its own nature, flows
equably __without__ __relation__ __to__ __anything__
__external__, . . .." Also, "absolute space, __in__ __its__ __own__
__nature__, __without__ __relation__ __to__ __anything__
__external__, remains always similar and immovable." [Our emphasis]

6. __Absolute__ __space__ __and__ __absolute__ __time__
__are__ __mind__ __independent__.

It is clear from the comments on absolute space and time in the Scholium that
just as absolute space and time are not dependent for their existence on the
existence of objects or processes, so they are not dependent for their
existence on their being __thought__ by some mind.

7. __Absolute__ __space__ __and__ __absolute__ __time__
__are__ __immutable__.

With respect to absolute time, immutability involves two aspects: (l) absolute
time flows __equably__. What this amounts to is that absolute time is
"ticking off" at a regular rate so that two __identical__ processes which
occur over different stretches of time take the __same__ amount of absolute
time to occur. We ignore, for the moment, the complications that arise if we
ask how one could __know__ this to be the case given that absolute temporal
intervals are, by Newton's own definition, inaccessible to measurement; and,
(2) the parts "of duration," i.e., individual moments, are fixed in their
relative order to one another in some intrinsic way. What this means is that,
if a set of successive events (each of which is occurring at a different
instant of absolute time) were ordered from earlier to later, then this order
from earlier to later is unique. This may seem so obviously true about physical
time that even to say it is to invite confusion, but, in fact, the Special and
General Theories of Relativity entail that it is false. It is, therefore, a
significant feature of the Newtonian view.

In fact, it is possible, on Newton's theory, to define a __Universal__
__Time__. By this we mean a time which, if it could be measured, would be
valid for all observers wherever they might be in the universe. If any one
observer A were capable of determining that a certain duration of absolute time
had elapsed, then any other observer B who could make a similar determination
would come up with exactly the same values for the elapsed duration of the
interval in question. Of course, Newton admits that such direct measurement is
not in practice possible. Even if there were a clock whose motion was
sufficiently equable to serve
as a measure of the elapsed absolute duration, there is no way that anyone
could know this to be the case. Nevertheless, all processes which take a
certain amount of time as measured by (against) some other process, also take
up a certain amount of absolute time. This amount of absolute time is the same
for any given process for all observers.

In addition, the immutability of absolute time means that it is possible to define an absolute Now. Since the moments of absolute times are fixed and immutable, it follows that when A says "now," she singles out a particular definite moment of absolute time. This particular moment (call it "N"~ is the same for all observers in all places everywhere in the universe. If asked, any one of them would (were they capable of making the required measurements) agree that A's utterance occurred at absolute time N.

One of the consequences of this is that, for a Newtonian, it is possible to uniquely determine whether two events are or are not simultaneous, no matter how far apart in space they may occur. Two such events are simultaneous if and only if they occur at the same instant of absolute time.

We can construct the following model for such a Newtonian universe. Each moment in absolute time defines what we may call a "simultaneity slice." A simultaneity slice is a set of events and places all considered at the same moment. If we imagine a camera taking a sequence of instantaneous flash pictures of a process, then each picture would be analogous to a simultaneity slice of a small portion of the universe. Now imagine that we have a camera capable of taking an instantaneous 3-dimensional holographic snapshot of the entire universe. Each snapshot would be a picture of such a simultaneity slice. If we imagine two observers, at different parts of the universe, each possessing such a cameras will agree exactly for any given moment, i.e., they can be perfectly superimposed on one another. Thus, when Newton says that the spatial universe endures forever through time, we can understand this from the picture we have just drawn to mean that corresponding to each moment of absolute time there is a unique portrait of the universe as a whole. The succession of portraits represents the unique history of the universe (for a further discussion of this model, see chapter 9). All of this may appear trivial and commonplace. However, the ~pecial and General Theories of Relativity deny the possible existence of such portraits and are incompatible with the possibility of an Absolute Now or a Universal Time function.

In saying that absolute space was immutable, Newton was especially concerned
about the immobility of parts or positions of absolute space. In particular, he
was concerned with what took to be the disastrous implications of the Cartesian
view that, in __identifying__ space with the bodies that (seemingly~ occupy
space, led to the view that positions or parts of space could move with respect
to one another. Since Newton held that one could only understand motion in
terms of objects being first in __one__ place and then in __another__, if
places themselves move, then after a body has moved from one place to another,
we would no longer be in a position to say from __where__ the motion had
started since the original place from which the body moved would have, in the
mean time, moved to where the object was now as well (compare the similar
argument by Aristotle discussed in chapter 5, pp. llff). For Newton, this is
just to deny that objects move and to make their motion unintelligible. But,
objects do move and their motion is intelligible, therefore, the Cartesian view
is false. The immutability of absolute space allows absolute space to serve as
a framework in terms of which motions can be defined and understood. Without
such a fixed framework, Newton felt that the concept of motion was
incoherenL.
Although he does not say so and he is notparticularly aware of any problems
here, Newton's claim that space is immutable can be interpreted in hindsight to
entail that, for Newton, the __geometrical__ structure of the parts of space
remains constant through time. Since Newton was not alive to the possibility
that the geometrical structure of physical space could be other than Euclidean,
he could not very well have been concerned about the possibility that the
geometrical structure of physical space could change. But, the Special Theory
of Relativity naturally suggests that the structure of the 4-dimensional
complex of space and time together is __not__ Euclidean, and certain models
of the General Theory of Relativity allow for the possibility that the
geometrical structure of space itself changes from moment to moment. Thus,
Relativity Theory gives a sense to the claim that space and time (for the
structure of time as well need not be the same from moment to moment or place
to place according to the general theory) are mutable. (For a discussion of
alternative geometries in the context of different theories of space and time,
see van Fraassen, 1970, chapter IV.)

8. __Absolute__ __space__ __and__ __absolute__ __time__
__are__ __causally__ __inert__.

For Newton, space and time are container-like frameworks within which the actions and events of the world take place. But, space and time themselves do not causally interact with the material or mechanical processes that go on in the world. Thus, for Newton, for example, the gravitational attraction of two bodies for each other does not affect and is unaffected by the structure of absolute space and absolute time. This is similar to the sense in which the structure of a rigid container is not affected by any objects placed in it or processes that occur in it. Again, this may appear to be a trivial and commonplace fact about physical space but it is not since it will have to be abandoned when we move to the General Theory of Relativity.

9. __Absolute__ __space__ __is__ __a__ __void__.

For Newton, absolute space exists independently of the existence of any bodies in it. Therefore, it is possible for there to be on Newton's view, empty spaces. This again is a non-trivial claim since it is denied by both Descartes and Leibniz as well as by Aristotle.

It means that not only absolute space but most of the relative space that is
observable by us, e.g., the space between the planets is empty too. This
created problems for Newton, as we have seen since it meant that the
gravitational force acting between two planets, for example, had to act over a
distance wlth no intermediate medium. As noted above, Newton was always unhappy
about the idea of action-at-a-distance. In the Optics, he goes so far as to
suggest that there might be an omnipresent a__ether__ to serve as the medium
for the gravitational force (not to be confused with the l9th century
__aether__ which was invented for the transmission of light rays. Newton had
no need for such an optical aether because, on his view, light was composed of
a stream of particles which could just as easily be shot across a void as
through a luminiferous aether). Newton never adequately resolved this problem
nor did any subsequent Newtonian. What was needed was a new way of thinking
about gravity which finds its first full expression in Einstein's General
Theory of Relativity.

This completes our survey of the general characteristics of Newtonian space and
time. We turn now to a postponed question, namely, why Newton felt it
obligatory to distinguish between the relative spaces and times which were
measurable and observable by human beings and absolute space and absolute time
which were not. We have already noted that Newton felt that without such
concepts the concept of motion would be unintelligible, but, in addition, he
had a number of experimental arguments which are often interpreted as intended,
by Newton, to support his absolutistic position. III. __Newton's__
__Arguments__ __for__ __the__ __Existence__ __Of__
__Absolute__ __Space__ __and__ __Absolute__ __Time.__

A.__ The Existence of Absolute Time__.

With respect to the distinction between absolute time and relative time, Newton has the following to say: Absolute time, in astronomy, is distinguished from relative, by the equation or the correction of the apparent time. For the natural days are truly unequal, though they are commonly considered as equal, and used for a measure of time; astronomers corrected this inequality that they may measure the celestial motions by a more accurate time. It may be, that there is no such thing as an equable motion, whereby time may be accurately measured. All motions may be accelerated and regarded, but the flowing of absolute time is not liable to any change. . . The duration or perseverance of the existence of things remains the same, whether the motions are swift or slow, or none at all: and therefore this duration ought to be distinguished from what are only sensible measures thereof; from which we deduce it, by means of the astronomical equation. The necessity of this equation, for determining the times of the phenomenon, is evinced as well from the experiments of the pendulum clock, as by the eclipses of the satellites of Jupiter. (Newton, 1962, PP- 7-8)

Newton is here alluding to the well known fact that different processes which might be used to measure relative times do not always agree with one another on the times that they determine. This was pointed out earlier in our discussion of Aristotle (pp. 5-47ff.) and was also mentioned in the context of our discussion of the implications of the Copernican revolution (pp. 6-67ff). We noted that there are a number of astronomical phenomena which are periodical and might be used to define or measure time. Unfortunately, the scales determined by these different measures do not remain synchronized with one another. Thus, one defines a solar day as the time interval from one local noon to the next. On the other hand, the motion of the moon can be used to define time as well. The natural unit is the time between two appearances of the new moon which we might call the lunar month. Since the background stars are, for all practical purposes fixed in relation to one another, we can define a sidereal day by fixing on a star which is always present in the night sky and defining the sidereal day to be the time it takes the star to reappear in a given position in the sky from night to night. The problem which gives rise to what Newton calls "the equation of time" is that these three measurements do not agree with one another. If we take the time to be measured by the solar day, then the time for a lunar month varies according to the time of year. Conversely, if we take the lunar month as our standard, then it turns out that the number of solar days varies from month to month. Similarly, a given number of solar days does not equal an equivalent number of sidereal days and as time goes on, the discrepancy gets larger and larger. The discrepancies between the solar, lunar and sidereal standards obey a mathematical equation (which is the equation of time) and by using it Newton thought one could get closer to the "true" time of which each was an imperfect measure.

In the early part of the 17th century, Galilei had suggested using the periodic motion of a pendulum as a measure of time. If this is done, then, because the earth in moving around the sun, is not moving in a perfectly circular orbit, and has its axis tilted with respect to the plane defined by the earth and the sun, the solar days are not equal in length at different times of the year.

In 1676, the Dutch physicist Ole Roemer had made observations of Jupiter and its moons and had measured the periods of the Jovian moons by observing them being eclipsed by Jupiter (the referent of Newton's remark about "eclipses of the satellites of Jupiter. Roemer also used his measurements to conclude that the speed of light was finite; see the discussion in chapter 10). If one takes the natural solar day as the standard of time, then it turns out that the periods of Jupiter's moons vary according to the time of year which they are observed. But, this contradicts one of the fundamental assumptions of Newtonian mechanics, namely, that the motion of one body in a stable orbit around another is due to the presence of a more or less constant gravitational force. If the times required to complete one period varied from one time of year to another, this would suggest that there must be some further force in addition to the gravitational force which was perturbing the motion of the moons. However, when the periods of the moons were compared with the time as measured by a pendulum clock, no such discrepancies between periods was observed. This suggests that the pendulum clock is a more "regular" standard of temporal measurement than the apparent motion of the sun, at least for the purposes of physics.

Newton's conclusion from these discrepancies was that each of these motions or processes gave only an imperfect measure of some "true" time. This "true" time could be calculated by means of the mathematical equation he called the equation of time. However, as we pointed out in our earlier discussion of the same subject (pp. 5--47ff.), there is no need to assume nor any justification for us to conclude that any such measure would be "more accurate" than any other measure. The fact that we can define a "mean solar day" or something of the sort just means that we have replaced one standard of measurement by another. But the conclusion that Newton draws, which is unwarranted, is that such a standard is, in effect, a more equable measure of absolute time. Newton was treating what is now understood to be a convention as a matter of fact, namely, what the standard for measuring time should be. We now think there is open to us to choose any number of different processes or motions as potential standards. We choose our standards usually on the basis of convenience: in this case, the physics turns out simpler if we choose the pendulum clock over the other choices (and even more convenient for modern physics if we choose the period of some radiating atom).

B. The Existence of Absolute Space.

With respect to the distinction between absolute space and relative spaces,
Newton offers two arguments in the __Principia__. One is based on an
interpretation of the results of an experiment which Newton conducted and which
can be conducted in any laboratory. It is generally known as "the bucket
argument." The other argument is a "thought experiment" which we will call "the
two globe experiment." It is a thought experiment insofar as Newton asks us to
imagine that would be the outcome of an experiment which cannot, in practice,
be performed.

(i) Newton's Bucket Experiment.

The bucket experiment is described by Newton as follows: The effects which distinguish absolute from relative motion are, the forces of receding from the axis of the circular motion. For 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 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 in a 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, gradually communicating its motion to the water, will make it begin sensibly to revolve, and recede by little and little from the middle, and ascend to the sides 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 least, performing its revolutions in the same times with the vessel, it becomes relatively at rest in it. This ascent of the water shows its endeavor to recede from the axis of its motion; and the true and absolute motion of the water, which is here directly contrary to the relative, becomes known, and may be measured by this endeavor. At first, when the relative motion of the water was greatest, it produced no endeavor to recede from the axis; the water showed no tendency to the circumference, nor any ascent towards the side of the vessel, but remained of a plain surface, and therefore its true and circular motion had not yet begun. But afterwards, when the relative motion of the water had decreased, the ascent thereof towards the sides of the vessel proved its endeavor to recede from the axis; and this endeavor showed the real circular motion of the water continually increasing, till it had acquired its greatest quality, when the water rested relatively in the vessel. And therefore this endeavor does not depend upon any translation of the water in respect of the ambient bodies, nor can true circular motion be defined by such translation. There is only one real circular motion of any one revolving body, corresponding to only one power of endeavoring to recede from its axis of motion, as 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 all together destitute of any real effect, any otherwise then they may perhaps partake of that one only true motion. And therefore in their system who supposes that our heavens revolving below the sphere of the fixed stars, carry the planets along with them; to several parts of these heavens and the planets, which are indeed relatively at rest in their heavens do yet really move. For they range their positions one to another (which never happens to bodies truly at rest), and being carried together with their heavens,partake of their motions, and as parts of revolving wholes, endeavor to recede from the axis of their motions. (Newton, 1962, pp. lOf.)

There are at least two interpretations of what Newton is trying to accomplish
in this passage. One, which we will call the standard interpretation has been
given its fullest expression in Nagel (1962). It construes the passage as an
attempt by Newton to experimentally prove the existence of absolute space. The
second, recently put forward by Ronald Laymon (1978), urges that Newton is not
trying to prove that absolute space exists, but rather is illustrating one of
the consequences of __assuming__ that absolute space does exist, an
assumption Laymond thinks Newton makes on other grounds. We examine both in
turn. First the standard interpretation.

a. __The__ __Standard__ __Interpretation__

On the standard interpretation the essence of Newton's argument is as follows.
Absolute time and absolute space, in and of themselves, are not detectable. But
there are detectable phenomena which we may use to infer the existence of
absolute space and absolute time. The "true and absolute circular motion of the
water" serves for Newton as an indication of the existence
of __absolute__ acceleration, i.e., acceleration of bodies with respect to
absolute space. However, the existence of __absolute__ accelerations implies
the existence of __absolute__ __velocities__ even though these latter
(absolute velocities) are not directly experimentally detectable. The existence
of absolute velocities follows from the fact that "acceleration" is defined as
the rate of change of a velocity. Thus, if a body has an __absolute__
__acceleration__, then that __acceleration__ must be the rate of change
of some __abs__ol__ute__ __velocity__. But, if there is an
__absolute__ __velocity,__ then the body in question must be moving with
respect to absolute space. Therefore, absolute space must exist. This (alleged)
inference by Newton has been vigorously challenged. On the standard
interpretation, Newton is guilty of an elementary logical blunder. We turn to a
more detailed analysis of the bucket experiment the stages of which are
diagrammed below. [See figure 7-1]

The standard interpretation of Newton's bucket experiment breaks it down into five stages as the accompanying diagram indicates. Newton's description stops at Stage 3, Stages 4 and 5 are added to give the experiment a certain symmetry. In Stage 3, the water and the bucket are at rest relative to one another but are both rotating with respect to the laboratory. Suppose, at this point, the bucket is abruptly stopped. It is now at rest with respect to the laboratory and the water which is contained therein continues to rotate with respect to both the bucket and the laboratory. Gradually (actually, fairly quickly) the surface of the water will resume its earlier plain shape as the rotational motion of the water decreases. The final stage, Stage 5, will put the experimental apparatus back into the initial condition with the water and the bucket both at rest with respect to one another and with respect to the surrounding laboratory. The analysis proceeds as follows. Compare Stage 1 with

Stage 1: Water and bucket at rest. Stage 2: The water and the

bucket are acceler-

ating with respect

to one another.

Stage 3: The water and the bucket are Stage 4: The water and bucket are

at rest relative to one another. in relative acceleration,

Both are rotating with respect when the bucket is stopped

to the laboratory. I abruptly.

1~ F~9 1- l'

Stage 5: The water and bucket come to rest with respect to each other and with
respect to the laboratory. with Stage 3 and Stage 2 with Stage 4. In stages 1 and
3, the water and
the bucket are relatively at rest with respect to one another. However, the
shape of the water surface in Stage 1 is planer, but in Stage 3 it is concave.
Similarly, in Stages 2 and 4, the water and the bucket are accelerating with
respect to one another. Again, in Stage 2 the surface of the water is planer,
while in Stage 4 the surface is concave. The conclusion to be drawn is that the
shape of the water surface is __independent__ of the relative state of
motion of the water with respect to the bucket. What does this mean?

The paraboloidal surface of the water in Stages 3 and 4 is the result of a
__deformation__ of the original water surface in Stage 1. But, bodies do not
spontaneously deform themselves. They undergo deformations only in response to
the presence of forces acting upon them. Therefore, the deformation of the
water in Stages 3 and 4 is an indication of the presence of some __forces__
acting on the water during Stages 3 and 4 that were not present during Stages
1, 2 and 5. Newton's SECOND LAW OF MOTION tells us that forces indicate the
presence of __accelerated__ motion. Since the water surface is deformed, the
water must be accelerating with respect to something. But, with respect to what
something is the water being accelerated? There seem to be three possible
alternatives:

(l) The water is accelerating with respect to the bucket.

(2) The water is being accelerated by some objects at a distance from the experimental apparatus, e.g., the fixed walls of the laboratory or the fixed stars.

(3) The water is rotating with respect to absolute space.

(l) is ruled out by the data of the experiment. Stage 3 indicates that the
surface of the water is deformed even when the water and the bucket are
relatively at rest with respect to one another. Thus, the deformation of the
water surface cannot be caused by the relative acceleration of the water with
respect to the bucket.
(2) is not ruled out by the data of the experiment but is ruled out by an
assumption that Newton made about the __inertia__ of bodies. The inertia of
a body is a measure of its tendency to resist changes in its state of motion
(recall Newton's FIRST LAW). The receding of the water from the rotating axis
of the bucket is an inertial effect, that is, its magnitude is a function of
the inertia of the contained water. Newton assumed that the inertia of a body
was an __intrinsic__ non-relational property that the body possessed. As
such, any effects due to it could __not__ be a function of any relationship
between the body in question and any other bodies in the universe. Thus,
alternative (2), which asserts that the inertial effects of the motion of the
water __are__ a function of its relationship to distant bodies, is ruled
out.

This leaves (3) as the only viable alternative: therefore, absolute space must exist.

Such is a brief outline of the standard interpretation of Newton's bucket
experiment (for more details see Nagel, 1962, chapter 8 ). The elementary
logical blunder which Newton committed on this view is to think that
alternative (3) __followed__ from the experimental result. In fact, it
follows only on the additional assumption that rules out (2). But this
assumption does not follow from the experimental result and can be (and was:
see chapter 8) rejected by Ernst Mach in the late 19th century.

b. __Laymon's__ __Interpretation__

Ronald Laymon has recently (1978) published a paper challenging the standard interpretation on a number of grounds and proposing an alternative view of what Newton was up to with the bucket.

First of all, the standard interpretation, as we have seen, commits us to
saying that Newton made a simple logical mistake. While it is not uncommon for
great men to make mistakes, the mistake Newton is supposed to have committed is
so obvious that it is hard to see how Newton could have failed to see that he was doing so. Perhaps then he did not make a mistake, in which case the
standard interpretation of what Newton was up to might be wrong. In fact,
Laymon argues that it is wrong. The experiment was __not__ designed to prove
the existence of absolute space at all, according to Laymon, because Newton was
already committed to the existence of absolute space on other (primarily
theological) grounds. What, then, was the point of the discussion? The
experiment, Laymon says, was designed as an __illustration__ of one effect
of accepting the doctrine of absolute space. At best, it can be construed as an
indirect argument supporting the existence of an absolute space which as been
postulated on other grounds. The structure of the argument is alleged to be as
follows:

(l) The existence of absolute space entails the bucket result.

(2) The bucket result holds (experimental fact). Therefore,

(3) The existence of absolute space is confirmed.

Such an argument, which is typical of the inductive experimental reasoning of science, does not, of course, prove that absolute space exists because it does not rule out the possibility that the bucket result might be due to some other se of circumstances (e.g., the circumstances outlined in alternative (2) of the standard interpretation). Newton is construed as offering a legitimate confirmatory argument and not as committing a blunder of any kind.

As supporting evidence for his interpretation, Laymon points out that a careful
reading of Newton's text shows that Stages 4 and ~ of the standard
interpretation are embellishments without textual support. In fact, Laymon
argues, the passage is primarily designed as an argument to refute Descartes'
theory of true and philosophical motion." Descartes' definition of "true and
philosophical motion" is "the translation of a piece of matter (a body) from
the neighborhood of the bodies immediately touching it, these being regarded as
at rest, to the neighborhood of others." (Laymon, 1978, p. 404) The bucket
experiment shows that such a relational view of motion is
__not__ __adequate__ to the __dynamical__ facts revealed by the
experimental result. The argument basically is as follows:

(a) The force acting on a body is proportional to the "real circular motion" of the body. [Both Descartes and Newton accept this premise, cf., Laymon, 1978, pp. 405ff.]

(b~ The force acting on the water is proportional to the curvature (deformation) of the water surface. [follows from Newton's FIRST and SFCOND LAWS]

(c) The curvature of the water surface is __inversely__ __proportional__
to the relative local motion of the water with respect to the bucket. [The
Bucket Result]

Therefore,

(d) The real motion of the water is __inversely__ __proportional__ to the
relative local motion of the water with respect to the bucket.

This follows from (a), (b) and (c).

The conclusion (d) contradicts the Cartesian view, implicit in Descartes'
definition of true and philosophical motion, that the real motion of the water
is __directly__ __proportional__ to the relative local motion of the
water with respect to the bucket.

Laymon's interpretation is plausible and a close reading of the text does seem
to support his view. His interpretation has the virtue that it absolves Newton
of the crime of committing a logical fallacy. One point against it is the
consideration that Newton does not specifically name Descartes as his foil. Why
not, if indeed the bucket is designed as an argument against the Cartesian
position? In fact, Descartes' name appears nowhere in the __Principia__. The
evidence that Descartes is the target rests on indirect evidence: the wording
of the bucket passage is reminiscent of Descartes; and, Newton's notebooks are
filled with speculations and queries explicitly about the Cartesian system (see
Laymon, 1978, for a discussion of the evidence). (ii) Newton's Globe Experiment
We turn now to Newton's thought experiment involving the two globes. Newton
writes: It is indeed a matter of great difficulty to discover, and effectually
to distinguish, the true [absolute] motions of particular bodies from the
apparent [relative]; because the parts of that immovable [absolute] space, in
which those motions are performed, do by no means come under the observation of
our senses. Yet the thing is not altogether desperate; for we have some
arguments to guide us, partly from the apparent motions, which are the
differences of the true motions; partly from the forces, which are causes and
effects of the true motions. For instance, if two globes, kept at a given
distance, one from the other by means of a cord that connects them, were
revolved about their common center of gravity, we might, from the tension of
the cord discover the endeavor of the globes to recede from the axis of their
motion, and from thence we might compute the quantity of their circular
motions. And then if any equal forces should be impressed at once on the
alternate basis of the globes to augment or diminish their circular motion,
from the increase or the decrease of the tension of the cord, we might infer
the increment or decrement of their motions, and hence would be found on what
basis those forces ought to be impressed, that the motions of the globes might
be most augmented; that is, we
might discover their hindmost faces, or those which, in the circular motion do
follow. And thus we might find both the quantity and the determination of the
circular motion, __even__ __in__ __an__ __immense__ __vacuum,__
__where__ __there__ __was__ __nothing__ __external__ __or__
__sensible__ __with__ __which__ __the__ __globes__ __could__
__be__ __compared.__ __But__ __now__, __if__ __in__
__that__ __space__ __some__ __remote__ __bodies__ __were__
__placed__ __that__ __kept__ __always__ __a__ __given__
__position__ __one__ __to__ __another,__ __as__ __the__
__fixed__ __stars____do__ __in__ __our__ __regions,__ __we__
__would__ __not__ __indeed__ __determine__ __it__ __from__
__the__ __relative__ __translations__ __of__ __globes__
__among__ __those__ __bodies,__ whether the motion did belong to the
globes or to the bodies. But if we observed the cord, and found that its
tension was __that__ __very__ __tension__ __which__ __the__
__motions__ __of__ __the__ __globes__ __required__, __we__
__might__ __conclude__ __the__ __motion__ __to__ __be__
__in__ __the__ __globes,__ __and__ __the__ bodies to be at rest;
and then lastly, from the translation __of__ __the__ __globes__ __among__ __the__
__bodies,__ __we__ __should__ __find__ __the__
__determination__ of their motions. (Newton, 1962, p. 12 [our emphasis])

The underlined section of the cited material is the thought experiment. It is a thought experiment because it assumes that one could place two globes connected by a string in an immense vacuum and then place, at will, bodies at immense distances from them. Such an experiment is not possible to perform although we can think what the consequences might be if we could perform it. Figure 7-2 illustrates the essentials of the experiment. [See Figure 7-2]

In both cases, a pair of globes connected by a string is situated in a vacuum far from any other bodies. We may assume that a dynamometer (i.e., a device for measuring the tension in the strings) is attached to the strings halfway between the two globes. The globe of the outer circle we may take to be some configuration of distant objects. In case 1, the distant globe is rotating in a clockwise direction with respect to the pair of globes sitting in the center of the diagram Newton's Globes

Case 1a

F~g. ~

Case 2a

In case 2, the pair of globes in the center is rotating in a counterclockwise direction with respect to the distant globe. It is obvious that the relative motions represented by the two cases are exactly the same.

Merely by observing relative motions we cannot distinguish case 1 from case 2. However, Newton argues, there is difference between the two situations: in case 1 there is no measured tension in the string connecting the two globes; in case 2, there is a measured tension in the string. The tension in the string in case 2 is due, according to Newton, to the fact that, in case 2, the globes are "really" rotating whereas in case 1, they are only "apparently" rotating.

This situation seems analogous to the bucket case. We have two cases where the relative motions are the same but a force appears in one case but not in the other. The existence of this force is an indication that the globes in case 2 are accelerating with respect to something. By the design of the experiment, the only objects in the universe are the two globes with their connecting string and dynamometer and the configuration of distant objects. The accelerations of the two globes with respect to the distant configuration of objects is the same in both cases. Yet there is a force in case 2 but not in case 1. Therefore, the cause of the force cannot be the relative acceleration of the globes with respect to the configuration of distant objects. The only plausible option seems to be that the globes, in case 2, are rotating with respect to absolute space. In the light of Laymon's interpretationof the bucket experiment we should proceed with caution in trying to figure out what Newton is up to here. Laymon argues that the point of the two globes experiment is, again, to illustrate the consequences of accepting absolute space rather than to prove that absolute space exists (Laymon, 1978, p. 408ff). Laymon's view is that the globes experiment is intended to be a summary of three different means of distinguishing true from apparent motion (by properties, causes, and effects) which Newton had earlier discussed in the Scholium. This is certainly more plausible than the standard interpretation which sees the globe experiment to be an attempt to demonstrate the existence of absolute space. This view has Newton banking rather heavily on a thought experiment the outcome of which is not clear. If the only objects in the universe were the two rotating globes and their connecting string Newton's argument entails that when the globes are "really" rotating as opposed to being at rest (in this case there is no relative motion in either case), then a tension exists in the string. But, asks, the standard interpretation, how can Newton know this unless he has already assumed that absolute space exists? If he has already done so, then this is a bad argument for it. On Laymon's view, it is not a bad argument for the existence of absolute space because it is not an argument for the existence of absolute space at all (See the discussions of Mach in the next chapter.).

(iii) A Theoretical Consideration

The discussion of the bucket and globes experiments has led us to the view that Newton must (on pain of logical error) have had other grounds for holding that absolute space and time exist. Laymon has suggested, and our earlier discussion supports the view, that religious considerations played a role in Newton's adopting his absolutist position.

Here we want to consider the possibility that Newton might have thought that
the existence of absolute space and time was a theoretical requirement of his
mechanics. Certainly, if Laymon's argument is correct, Newton thought that the
concept of motion was unintelligible without some notion of absolute space and
time. Now consider Newton's FIRST LAW, the Law of Inerita. That law, recall,
stated that "every body continues in its state of rest or of uniform motion in
a right [straight] line, unless it is compelled to change that state by forces
impressed upon it." One question that disturbed later Newtonians
who were not as convinced of the existence of absolute space as was Newton was
how to understand the phrase "in a right line." particle paths are not straight
in and of themselves, on the Newtonian view, but are straight or curved only in
relation to some frame of reference. The problem for the Newtonians was what
frame of reference was to be used in formulating the fundamental principle of
inertia? Newton himself does not seem to have worried too much about it. The
assumption is that he did not worry about it because he assumed that absolute
space and absolute time existed and that one could, in principle at least,
think of the inertial properties of objects as manifesting themselves in
relationship to absolute space and absolute time. Thus, the Law of Inertia is
taken to assert that every body will continue in its state of rest or uniform
motion in a straight line with respect to __absolute__ __space__
__and__ __absolute__ __time.__

As a matter of fact, the concepts of absolute space and absolute time do not play an active role in the application of Newtonian principles to physical systems (We should note that if the standard interpretation of the bucket experiment commits Newton to a logical error, this interpretation commits him to an error in physical reasoning). Perhaps this partly explains why Newtonian physics remained secure until the characteristic equations which connect one observer's description of a physical system with another's were themselves challenged at the end of the l9th century.

Absolute positions and absolute moments of time, Newton declared, were
undetectable, and absolute motion was __almost__ __completely__
undetectable. As long as one is dealing with __unaccelerated__ motions, only
relative motions, positions and times can be detected. Newton argued that if
two observers are moving with constant velocity with respect to one another,
there is no way, in principle, that they can determine by a mechanical
experiment which of them is "really" moving. This is known as the
__Galileian__ __Principle__ __of__ Relativity (its first

statement can be found in Galilei's (1970, p. 186) The Principle gives rise
to what are called the __Galilean__ __Transformations__. Given a physical
situation as described by one observer, the transformations tell how an
observer who is moving with constant velocity with respect to the first wlll
describe the same situation.

Suth systems of observers moving with constant velocities (not necessarily the
same) with respect to one another are called __inertial__ observers and they
define what is known as an __inertial__ __frame__ __of__
__reference.__ Consider observers moving only in 1 dimension along the
x-axis.

o ~ r~

o x

Figure 7-3

The **unprimed observer we will consider to be at rest, **O' moves along ~he
x-axis with a velocity V relative to 0. Suppose, by some happy coincidence,
that the origin of the X-system coincides with the origin of the X'-system when
the clocks of the two observers read T ~ T' = 0. At some time t later, an event
occurs at a position p units from observer 0.

~x~

v

o ~ ~

F g~-~

According to 0, 0' has moved a distance - Vt down the x-axis. Of course, O'
does not think that __he__ has moved at all. He thinks O has moved -Vt'
units down the x'-axis. Both observers think __they__ are at rest and the
other has moved. The only thing they agree on is that they are separated by the
distance d= Vt. In general, if O sees an event p occur at some point x in the
O-frame of reference, O' will see the event occur at a point x' in the O'-frame
of reference, where x and x' are related by the equation: x = x' + Vt'

Recall that time is absolute in the Newtonian world view. This means that the time as measured by the O-clock (=t) and the time as measured by the O'-clock (=t') are related by the equation t = t'

The three equations

are the __Galilean__ __Transformations__ in one dimension. Actually, only
one of the first two is needed since the other can be derived from it using the
third. These equations can easily be generalized for motions in 3 dimensions
but we will not stop to pursue this now.

From these equations, one can easily derive equations which relate the velocity of an object as measured by O to the velocity of the same object as measured by 0'. The result is that if the velocity of an object A with respect to O is V, then the velocity of the object with respect to O' (=V') is given by V' = V + v where v is the velocity of O' with respect to 0. Similarly, if A is accelerating with respect to O with an acceleration a, then its acceleration as measured by O' (= a') is given by

This is the crucial result. Any acceleration seen by one inertial observer is
seen as exactly the same by any other inertial observer. But, in Newtonian
mechanics, forces are proportional to accelerations. Thus, the Galileian
transformations, characteristic of Newtonian mechanics, imply that any two
inertial observers will __agree__ on a description of the forces acting in
any given situation. This is what one means by saying that they "see" the same
physics. Because of this fact, if two observers are moving with constant
velocity with respect to one another, there is __no__ __way__ for them to
determine which observer is "really" moving. This is what one means by saying
that absolute motions are not detectable by __kinematic__ (motion)
considerations alone. Newton's arguments concerning absolute motions all
involve __dynamic__ (force) considerations. But, these latter effects are
equivalent for inertial observers who are said to be __Galilean__
__invariant__.

The net effect is that the system of inertial observers functions just like absolute space and time does. The principle of inertia holds for such observers but no absolute motions are detectable by them. The only hope for detecting absolute motion rests on finding some (non-mechanical) effect or force which will single out one group of inertial observers as being somehow distinguished from the rest (i.e., "really" at rest). In the l9th century it was discovered that the laws of electromagnetic radiation are not Galilean invariant. Experiments were set up to detect absolute motion but they were not successful. The implications of these results led to the rejection of Newton's concepts of absolute space and absolute time. (See chapter 9 for a discussion of the role of optical phenomena in the downfall of Newtonian mechanics.)

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