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TESTING NEWTON, THEN AND NOW
THE LOGIC OF EVIDENCE IN GRAVITY RESEARCH
FROM NEWTON THROUGH THE 20TH CENTURY
George E. Smith
Dibner Institute for the History of Science
and Technology
Philosophy Department, Tufts University
WHY NOT SIMPLY HYPOTHESIS TESTING BY MEANS
OF DEDUCED PREDICTIONS?
HEMPEL’S PROVISO PROBLEM
• Deduced predictions in celestial mechanics presuppose
a proviso: no other (consequential) forces are at work.
• The only evidence for this proviso is close agreement
between the predictions and observation.
• But then a primary purpose of comparing deduced
predictions and observation is to answer the question,
Are other forces at work?
• How then is the theory of gravity tested in the process?
How did we first come to have high
quality evidence in any science?
Why was orbital astronomy so much better
at turning data into evidence after
Newton’s Principia than it was before?
OUTLINE
I.
Introduction: the issue
II.
The logic, as dictated by Newton’s Principia
III.
How this logic played out after the Principia
IV.
A.
“Then” – complications that obscure the logic
B.
“Now” – in light of the perihelion of Mercury
Concluding remarks
“GRAVITY RESEARCH” THEN AND NOW
IN CELESTIAL MECHANICS:
What are the true motions – orbital and rotational – of
the planets, their satellites, and comets, and what forces
govern these motions?
IN PHYSICAL GEODESY:
What is the shape of the Earth, how does the gravitational
field surrounding it vary, and what distribution of density
within the Earth produces this field?
CALCULATING PLANETARY ORBITS — 1680
NEWTON’S EVIDENCE PROBLEM
IN THE PRINCIPIA
“By reason of the deviation of the Sun from the center of gravity,
the centripetal force does not always tend to that immobile center,
and hence the planets neither move exactly in ellipses nor revolve
twice in the same orbit. There are as many orbits of a planet as it
has revolutions, as in the motion of the Moon, and the orbit of any
one planet depends on the combined motion of all the planets, not
to mention the action of all these on each other. But to consider
simultaneously all these causes of motion and to define these
motions by exact laws admitting of easy calculation exceeds, if I
am not mistaken, the force of any human mind.”
Isaac Newton, ca. December 1684
(First published by Rouse Ball in 1893)
INFERRING LAWS OF FORCE FROM
PHENOMENA OF MOTION
Phenomena: Descriptions of regularities of motion that hold at least quam
proxime over a finite body of observations from a limited period of time
The planets swept out equal areas in equal times quam proxime with respect to the Sun
over the period from the 1580s to the 1680s.
Propositions, deduced from the laws of motion, of the form:
“If _ _ _ quam proxime, then …… quam proxime.”
If a body sweeps out equal areas in equal times quam proxime with respect to some point,
then the force governing its motion is directed quam proxime toward this point.
 Conclusions: Specifications of forces (central accelerations) that hold at
least quam proxime over the given finite body of observations
Therefore, the force governing the orbital motion of the planets, at least from the 1580s to
the 1680s, was directed quam proxime toward the Sun.
From Evidence that is Approximate to
A Law that is Taken to be Exact
Rule 3: Those qualities of bodies that cannot be intended and
remitted and that belong to all bodies on which experiments
can be made should be regarded as qualities of all bodies
universally.
Rule 4: In experimental philosophy, propositions gathered
from phenomena by induction should be regarded as either
exactly or very, very nearly true notwithstanding any contrary hypotheses, until yet other phenomena make such
propositions either more exact or liable to exceptions.
This rule should be followed so that arguments based on induction may
not be nullified by hypotheses.
PREREQUISITES FOR TAKING THE
THEORY OF GRAVITY AS EXACT
• The theory must identify specific conditions under which the
phenomena from which it was inferred would hold exactly
without restriction of time – e.g.
– The area rule would hold exactly in the absence of forces from other
orbiting bodies
– The orbits would be perfectly stationary were it not for perturbing
forces from other orbiting bodies
• The theory must identify a specific configuration in which the
macroscopic variation of gravity about a body would result
from the microstructure of the body – e.g.
– Gravity would vary exactly as the inverse-square around a body were
it a sphere with a spherically symmetric distribution of density
TAKING THE THEORY TO BE EXACT
THE PRIMARY IMPLICATION
Every systematic discrepancy
between observation and any
theoretically deduced result
ought to stem from a physical
source not taken into account
in the theoretical deduction
– a further density variation
– a further celestial force
THE NEWTONIAN APPROACH
CONTINUING EVIDENCE
•
Taking the law of gravity to hold exactly was a research strategy,
adopted in response to the complexity of the true planetary motions.
•
Deductions of planetary motions etc. are “Newtonian” idealizations:
approximations that, according to theory, would hold exactly in
certain specifiable circumstances -- in particular, in the absence of
further forces or density variations.
•
The upshot of comparing calculated and observed orbital motions is
to shift the focus of ongoing research onto systematic discrepancies,
asking in a sequence of successive approximations, what further
forces or density variations are at work?
•
Theory thus becomes, first and foremost, not an explanation (or even
a representation) of known phenomena, but an instrument in ongoing
research, revealing new “second-order” phenomena that can provide
a basis for continuing testing of the theory.
THE LOGIC OF THEORY TESTING
• The theory requires that every deviation from any “Newtonian”
idealization be physically significant – i.e. every deviation must
result from some further force or density variation.
• Basic Testing: pin down sources of the discrepancies and confirm
they are robust and physically significant (within the context of the
theory) while achieving progressively smaller discrepancies between
(idealized) calculation and observation.
• Ramified Testing: keep incorporating previously identified physical
sources of second-order phenomena into the (idealized) calculation,
thereby progressively constraining the freedom to pursue physical
sources for new second-order phenomena that then emerge.
• The continuing evidence lies not merely in the aggregate of the
individual comparisons with observation, but also in the history of
the development of the sequence of successive approximations.
NEPTUNE AS AN EXAMPLE OF
“PHYSICAL SIGNIFICANCE”
seconds of arc
THE “GREAT INEQUALITY” AS A
MORE TYPICAL EXAMPLE
minutes of arc
OUTLINE
I.
Introduction: the issue
II.
The logic, as dictated by Newton’s Principia
III.
How this logic played out after the Principia
IV.
A.
“Then” – complications obscuring the logic
B.
“Now” – in light of the perihelion of Mercury
Concluding remarks
Second-Order Phenomena Often Underdetermine
Their Physical Source
Example
Example
Deviation of surface gravity from
Newton’s ideal variation implies
the value of (C-A)/Ma2 and hence
a correction to the difference (C-A)
in the Earth’s moments of inertia,
and the lunar-solar precession
implies the value of (C-A)/C and
hence a correction to the polar
moment C; these two corrected
values constrain the variation (r)
of density inside the Earth, but they
do not suffice to determine (r) .
RESPONDING TO UNDERDETERMINATION
20TH CENTURY DETERMINATION OF (r)
density
density
core-mantle
boundary
ROBUSTNESS OF PHYSICAL SOURCES
Examples
• Mass of Moon inferred from lunar
nutation supported by calculated
tides and lunar-solar precession
• Mass of Venus inferred from a
particular inequality in the motion
of Mars supported by calculated
perturbations of Mercury, Earth,
and Mars
• Gravity fields of Jupiter and Saturn
supported by variations in period of
Halley’s comet
VARIETIES OF EVIDENTIAL LOGIC FOR
ROBUSTNESS
• Hypothetico-deductive: confirmed deduced predictions
– e.g. date of maximum for Great Inequality
• Glymour boot-strapping: confirmed deduced predictions
– e.g. inferred masses of planets
• Same effect, same cause (common origin inference)
– e.g. fluctuations in rotation of Earth
• Agreeing measurements
– e.g. masses, moments of inertia
• From success of research predicated on claim
– e.g. Uranus motion anomaly after Jupiter and Saturn effects
PROBLEMS IN ISOLATING DISCORDANCES
“The motion of the [lunar] perigee
can be got [from observation] to
within about 500,000th of the
whole. None of the values hitherto
computed from theory agrees as
closely as this with the value
derived from observation. The
question then arises whether the
discrepancy should be attributed to
the fault of not having carried the
approximation far enough, or is
indicative of forces acting on the
moon which have not yet been
considered.”
G. W. Hill, 1875
Newcomb’s Discordances, 1895
• Mercury’s perihelion
 was 29 times probable error
• Venus’s nodes
 was 5 times probable error
• Mars’s perihelion
 was 3 times probable error
• Mercury’s eccentricity
 was 2 times probable error
ANOTHER EXAMPLE OF DIFFICULTY
Many professional lives have been dedicated to the long series of meridian circle (transit) observations of
the stars and planets throughout the past three centuries. These observations represent some of the most
accurate scientific measurements in existence before the advent of electronics. The numerous successes
arising from these instruments are certainly most impressive. However, as with all measurements, there is
a limit to the accuracy beyond which one cannot expect to extract valid information. There are many
cases where that limit has been exceeded; Planet X has surely been such a case.
THE MANY SOURCES OF DISCREPANCIES
In observations:
1. Simple error – “bad data”
2. Limits of precision
3. Systematic bias in instruments
4. Inadequate corrections for known
sources of systematic error, incl.
5. Imprecise fundamental constants
6. Not yet identified sources of
systematic error
In theoretical calculations:
1. Undetected calculation errors
2. Imprecise orbital elements
3. Imprecise planetary masses
4. Insufficiently converged
infinite-series calculations
5. Need for higher-order terms
6. Forces not taken into account
7. Gravitation theory wrong
“The ultimate goal of celestial mechanics is to resolve the great question whether
Newton’s law by itself accounts for all astronomical phenomena; the sole means of
doing so is to make observations as precise as possible and then to compare them with
the results of calculation. The calculation can only be approximate….”
Poincaré, 1892
“SECULAR” MOTION OF THE MOON
18th Century:
Acceleration in motion of Moon
announced by Halley (1693)
A physical source identified by
Laplace (1787):
Owing to perturbations
from gravity toward the
planets, eccentricity of
Earth’s orbit changing.
19th Century:
Adams finds that Laplace has
accounted for only half of the
“secular” motion (1854)
A further physical source: earth
is slowing from tidal friction
EXAMPLE OF SPECTACULAR SUCCESS
SPENCER JONES (1939)
• Residual discrepancies in the motions of Mercury, Venus, and Earth
correlate with unaccounted-for discrepancy in lunar motion
• Common cause => Earth’s rotation irregular (in more ways than one)
• Expose a still further systematic observation error, requiring correction:
– 1950: replace sidereal time with “ephemeris time”
This form of evidence can be very strong
• It is evidence aimed at the question of the physical exactness
of the theory
• The sequence of successive approximations leads to new
second-order phenomena of progressively smaller magnitude
• New second-order phenomena presuppose not only the theory
of gravity, but also previously identified physical sources of
earlier second-order phenomena, thereby constraining the
freedom to respond to these new phenomena
• Theory becomes entrenched from its sustained success in
exposing increasingly subtle details of the physical world
without having to backtrack and reject earlier discoveries
OUTLINE
I.
Introduction: the issue
II.
The logic, as dictated by Newton’s Principia
III.
How this logic played out after the Principia
IV.
A.
“Then” – complications obscuring the logic
B.
“Now” – in light of the perihelion of Mercury
Concluding remarks
INEXACTNESS EXPOSED:
THE PERIHELION OF MERCURY
“The secular variations already given
are derived from these same values of
the masses, the centennial motion of
the perihelion being increased by the
quantity
Dt = 43.″37
In order to represent the observed
motion. This quantity is the product
of the centennial mean motion by the
factor
0.000 000 0806”
PERIHELION OF MERCURY: CURRENT
FROM NEWTONIAN TO EINSTEINIAN GRAVITY
Discrepancy between Newtonian calculation and observation:
43´´.37 ± 2.1 ==> 43´´.11 ± 0.45
Increment from the Einsteinian calculation:
43´´. ==> 42´´.98
Newtonian gravity is the static, weak-field limit of Einsteinian!
A limit-case idealization
The orbital equation becomes, where μ = G(M+m), u = 1/r:
CONTINUITY OF EVIDENCE ACROSS
THE CONCEPTUAL DIVIDE
• 43´´ per century was a Newtonian second-order phenomenon
• From limit-case reasoning, evidence for Newtonian gravity
carried over, with minor qualifications, to Einsteinian
• Earlier evidential reasoning for Newtonian gravity, even
though requiring some qualifications, was not nullified
• Although curvature of space, gravitational time dilation, and
non-linear effects of mass on the metric were not previously
unidentified forces or density variations, they were still
physically intelligible in a Newtonian context
• Previously identified physical sources of Newtonian secondorder phenomena remained intact in Einsteinian
“… though the world does not change with a change
of paradigm, the scientist afterward works in a
different world…. I am convinced that we must learn
to make sense of statements that at least resemble
these.
Thomas S. Kuhn, SSR, p. 121
The continuity of evidence across the conceptual
divide between Newtonian and Einsteinian gravity
highlights an extremely important sense in which
the scientist afterward works in the same world.
PRIMARY CONCLUSIONS
• The most important evidence in classical gravitational research
came from the complexities of the actual motions and of the
gravitational fields surrounding bodies.
• This evidence consisted of success in pinning down physical
sources of deviations from “Newtonian” idealizations, in a
sequence of increasingly precise successive approximations.
• This evidence carried forward, continuously, across the transition from Newtonian to Einsteinian gravity and remains an
important source of continuing evidence today.
FIVE CORRELATIVE POINTS
• In response to a particular evidential problem
– May not carry over to other areas of research
• Theory primarily an instrument in ongoing research
– Not primarily serving to explain
• Two stages of evidence: initial, continuing
– Initial stage focuses on promise of the theory
• Importance of getting evidential logic right
– Hypothetico-deductive understates force of evidence
• Historical development itself a source of evidence
– Cumulative development versus back-tracking