Conclusion  page 273








Chapter 4.
Universal Gravitation  page 29
page 29

A close reading of the astronomical piece of mechanics of the solar system, serves to temper the conventional image of a vindicator of Newton's law of gravity against the evidence for decay of motion in the planets.

To mention, in some sense, that the trajectories of the planets are not stable, is an important issue in this book.
Anyway to doubt the importance of Newton's Law, is out of the question.

Nothing is said about apparent anomalies gathering toward a cosmic catastrophe; on the contrary, the state of the universe is assumed to be steady.

It is very difficult to make any claims about the whole of the universe or at cosmic scale.
Considering galaxies there can be catastrophes and galaxies don't have to be 'steady' (i.e. ever existing).

THe problem is not whether the phenomena can be deduced from the law of universal gravity, but how to do it.

There are two ways to use Newton's law:
 Considering individual objects large and small as point masses.
 Using densities of small areas.
The first method is in principle the most accurate and can be tested based on observations, but completely in practicle at large scale.
The second method can be used at large scale. The problem lies in the calculation of the densities involved.
The second method is also used to chalenge the evolution of the universe by using friedmann's equations. One of the assumptions is that the universe is homogeneous and isotropic. The problem is that the results are very different to test by means of observations.

Since that appeared to be impossible on a strict Newtonian construction of evidence, Laplace proposed modifying the law of gravity slightly.

The bigest problem with Newton's Law is the assumption that gravity acts instantaneous. To take into account that it does not is mathematical difficult and what is more important can not be done by hand. At least not accurate enough to test the modifications with observations.

He proceeded to try out the notion that gravity is a force propagated in time in stead of instantaneously.

I think we should give that a proper investigation.

Its quantity at a given point would then depend on the velocity of bodies as well as on their mass and distance.

IMO this sentence should be changed as such:
Its quantity at a given point would then depend on the mass of bodies involved as well as on their velocities and distances in the past. .

Even more interesting, the reasoning in this argument was not that of normal mathematical astronomy but was of the type that Laplace brought to physics in other, much later writings.

That the argumentation belongs to physics is obvious. See also page 34

As in other early papers Laplace's point of depature was an analysis by Lagrange.

Okay.

In 1774, Lagrange had argued that it was impossible to derive from the theory of gravity an equation for the acceleration of the mean motion of the moon giving values large enough to agree with observations.

It requires investigation why specific is written: 'large enough'. IMO the words 'large enough' can be omitted.

Lagrange then wondered whether resistance of the ether might be slowing the rotation of the earth enough to resolve the apparent discrepancy.

If based on observations it is established that earth rotation is slowing down, a simpler explanation lies in the tides.
See also: https://en.wikipedia.org/wiki/Lunar_distance_(astronomy)#Tidal_dissipation

Laplace for his part took the question to be involving the sufficiency of the law of gravity with implications for all of cosmology.

In principle that is okay.
page 30



Thereupon, he turned to the "principle of universal gravitation" in general, which he called the most incontestable truth of all of physical sciences.

Okay.

It rested, in his view, upon four distinct assumptions generally accepted among géometres, or persones doing exact science.

Okay.

Given their importance in marking out the main lines along which Laplace developed his celestial mechanics, we shall state them in the form of a close paraphrase:
 The force of attraction is directly proportionol to mass and inversely proportional to the square of the distance.
 The attractive force of a body is the resultant of the attraction of each of the parts that compose it.
 The force of gravity is propagated instantaneously.
 It acts in the same manner on bodies at rest and in motion

No comments here.

The inversesquare law of atraction came first

Okay




page 31



It is perhaps somewhat surprising to find that in the first stage of Laplace's astronomical work, his anlysis of the consequences following from the second of these assumptions, the principle that the attractive force exerted by a body is the result of the attractions exerted by all of its parts, took precedence over the theory of planetary motion.

Yes it is.







The tides were too complicated a phenomenon to tackle yet although Laplace attacked the soon afterward in a major calculation.

See also Chapter 7




page 32





So much for the longrange importance of the first two assumptions in Laplace's work; the later two, on the other hand, are interesting mainly for the discussion of them that he gave in the ensuing articles in the present memeoir, and it was here that he least resembled the celestial dogmatist for which he has sometimes be taken

Okay.

The third and fourth assumptions have adifferent standing form from the inversesquare law and the principle of attraction

Okay.

Anyone doing planetary astronomy would have thought it necessary to write down those two first two principles in the axiomatic structure of a treatise.

Okay.

Not so the young Laplace, who stated these assumptionsfor the sake of taking issue with them.

In fact very clever, by principle.

It is unreasonable, he observed immediately, to suppose that the power of attraction or any other force acting at a distance should be propagated instantaneously.

Correct.
What young Laplace should have done is to study a different case were this other force can be demonstrated. Maybe he did.

Our sense is rather that it should correspond in its passage to all the intervening points of space successively.

Tricky sentence.
The whole issue is that the (total) attractive force operated upon a body at present is a function of the distances (of all the bodies considered) in the past.

Even if communication should appear instantaneous, what happens in nature may well be different, "for it is infinitely far from the unobservable time of propagation to one that is absolutely nil"







page 33





Thus, he would try what followed from the supposition that gravitation does take place in time.

i.e. does not act instantaneous.


















page 34











It might be worthwhile, therefore, to try out the notion that gravitation takes time, which is to say the the term alpha*t/theta is not nil.

Okay

The velocity is then: theta/(alpha*T).

Okay

Laplace found that the velocity of the gravitational corpuscle is 7680000 times as great as the velocity of light.

Remarkable large velocity.








page 35







Such a mechanism would indeed explain the apparent acceleration of the moon, but Laplace had also found, by a method that he promised to give elswhere, that the rotation of the earth cannot be retarded by the friction of those winds in any detectable degree

Okay.

Thus we are left with the force of gravity, "astonishing" in its activity but finite in velocity

Interesting conclusion.














Chapter 7.
The Figure of the Earth and the Motion of the Seas  page 51
















Chapter 21.
Traité de mécanique céleste  page 184






page 195







Since the mean motion of the earth shows no change over a two thousand year span, Laplace calculated that the sun had not lost a twomillionth part of its substance in recorded history and the effect of the impact of light particles on the secular equation of the moon is undetectable.

Interesting.





Thus he ould have emerged full circle from his celestial mechanics, coming out just where he went in with the first calculation of the youthful probabilitygravitation memoir, except that now gravity is given a velocity of 1*10^9 times the speed of light, which is to say infinite

Again a remarkable result.
Anyway much larger than what General Relativity assumes, which is the same as the speed of light.






Conclusion  page 273



Belief in a fully determined universe was nonetheless a faith for being naturalistic rather than religious.

This raises immediate the first question: What means 'a fully determined universe'?
Suppose at this moment the whole of the universe is fully determined. At any next moment is the universe than still fully determined? How do you know that?
My honest opinion is that the whole concept does not make sense because we humans can only observe something about our nearby environment and with a very limited accuracy. The further away, the more in the past, the more inaccurate our observations become.

Laplace, even like Einstein after him, conceived the vision in his youth before setting out to prove it.

You can never prove something what is not clear, including the concept: to prove

A grand faith, and one not without works after all, determinism formed the expectations held to the exact sciences in the century and more following Laplace's death.

The issue is not determinism but stability. Stability is the concept that a cirtain process or situation exists for a long time and will continue to exist. If that is the cases the details can be studied, in time, observed, measured, calculated and be molded in a more mathematical notation or law. If that the cases, i.e. only for certain processes or certain situations, using the laws, we can predict the future for similar processes or situations. These cases we can call deterministic. Most cases are not, specific for example when human behaviour becomes involved. Such behaviour is often unpredictable.

It carried forward into Einstein's great vision of an underlying order in nature.

?





Already in the 1850s Le Verrier found that the values for eccentricity in Mécanique céleste hold good only if the mass of the planet in question forms a considerable part of the total mass of the planetary system, and that they were invalid for bodies as small as the earth.


Most recently, indeed very recently, it has been calculated in the light of chaos theory that the motions of the planets become unpredictable after some 100 million years.

There are in fact two issues involved:
 What is the physical state of the planets after 100 million years? Ofcourse we can make certain guesses, but no body knows this. Part of the problem is that during this 100 million years external objects will enter the solar system and, for example, will collide with the Sun or one of the planets, modifying their paths in an almost unpredictable manner.
This is a physical issue and has nothing to do with the chaos theory.
 A different issue is a simulation of the behaviour of our solar system on a PC.
One strategy is described in the book "Astronomical Algorithms" by Jean Meeus. This book uses analytical functions to calculate the positions of the planets. This is an excellent book to use if you want to calculate the positions in the year 2030, but becomes in accurate roughly "5000" years from now. This has also nothing to do with the chaos theory.
 A different strategy is to use Newton's Law. One important issue is computer accuracy (word size) and step size. Normaly this works very well, because it is very difficult to say that your simulation is wrong and does not match observations.
A typical case with causes a problem, is, if you want to simulate 4 identical objects moving in a flat circle around a central object, with the objects 90 degrees from each other. Such a symetrical simulation should stay stable for any period of time. This simulation requires accurate initial conditions in both position and speed. Suppose that the initial is a number like 10,0000000000 with 10 zero's behind the comma. What the simulation will show, based on word length (# of bits) is that very soon digit 10 (behind the comma) will change, then digit 9, then digit 8 etc until digit 1 changes. On the computer screen this becomes vissible because the objects will leave the circle shaped pattern. Specific now objects can be ejected or two objects can collide, which is not what you want.


page 274

He (Laplace) was an indefatigable calculator manipulating expressions that occupy many lines in argument that run for many pages.

It lies in the details. Okay.

The outcomes depend, virtually without exception, on approximate solutions to his problems and not on strict proofs.

When you want to calculate the position of the planets either you need a PC or when you have to do it by hand I expect the only way out are approximate solutions.
Anyway what are strict proofs? Progress in science starts when the new results, compared with the old results, are a better match with actual observations (or with an experiment).


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