8. Basic Concepts: The Law of Motion - page 2

8.2 Law 1 - page 22

Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it

Newton's First law discusses the influence of forces.

8.2 Law 2 - page 23

The change of motion is proportional to the motive force impressed; and is made in the right line in which that force is impressed.

Newton's Second law also discusses the influence of forces.

8.3 Law 3 - page 23

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

What is the sense of this law?
When I press against an object (on a surface) that object can stay at rest or move. I both cases I can press with the same force. The parameters involved are the weight of the object and the friction. Those are the deciding factors if the objects moves or stays at rest.

Whatever draws or presses another is as much drawn or presses by that other

This sentence does not say much.

If you press a stone with your finger, the finger is also pressed by the stone.

To claim that why I press a stone, the stone also presses me is tricky.
Newton's Third law can be considered in case where gravitation is involved, specific considering a rock on the surface of the earth. In that case two Forces of gravitation are at work: One force of the earth versus attracting the rock and a second force of the rock attracting the earth. Both forces are the same but in opposite directions. Read more on Page 33 and

page 26

Corollary IV

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

Next

In establishing Corolary IV first considers the case when the two 'bodies' in question are mass points mi (i=1,,,n) etc

This results in four equations identified as (6), (7), (8) and (9)

page 27

It is instructive to follow Newton's proof. He makes use of Lemma XXIII established later in book I:
Lemma XXIII If two given right lines, as AC, BD, terminating in given points A,B, are in a given ratio one to the other, and the right line CD, by which the indetermined points C, D are joined is cut in K in a given ratio; I say, that the point K will be placed in a given right line.

Next follows a drawing explaining Lemma XXIII
Next the sentence: We are required to find the locus of K, given the fixed point E, A and B and varying points C, D, and K satisfying the requirements:
What follows are three more equations (10), (11) and (12)

page 28

At page 28 follows equation (13)
The final sentence is:
that is, EH is determined by the initially given quantities, and therefore, remains constant as C, D and K vary as prescribed
Hence the locus of K is the straight-line HK prolonged parallel to EL.

What follows is the text:

Newton's proof (Using Lemma XXIII) proceeds as follows: etc

What follows is very interesting to read.

9. The scholium to the Laws of motion - page 30

page 31

It is equally certain that Galileo never enunciated the principle of inertia, and indeed could not have done so correctly, since 'horizontal' motion was for him always at the surface of the earth, equidistant from its centre, and therefore in reality circular and not rectilinear.

page 32

And thus the third Law, so far as it regards percussions (strikes?) and reflections is proved by a theory exactly agreeing with experience.

The question is in so far as experience can be used to prove something.
IMO science you follow the reverse part: First experiments (as may and possible) and as a result slowly a theory evolves. The next text assumes three objects: A,B and an obstacle

A brief demonstration:
Suppose an obstacle is interposed to hinder the meeting of any two bodies A,B, attracting one the other: then if either body, as A, is more attracted towards the other body B, than the other body B is towards the first body A, the obstacle will be more strongly urged by the pressure of the body A than by the pressure of the body B and therefore will not remain in equilibrium:

What is meant with equilibrium? Which objects are meant? I expect the obstacle.
The sentence continues with:

but the stronger pressure will prevail, and will make the system of two bodies, together with the obstacle, to move directly towards the parts on which B lies: and in free spaces, to go forwards in infinitum with a motion continually accelerated: which is absurd and contrary to the first Law.

To mention the first Law is not proper science, because the first Law can be wrong. The proper way to do science is try not to use other Laws.

page 33

So the gravitation between the Earth and its parts is mutual.

In the following text the Earth is divided in three parts: A small part N, created by the dividing plane going through the North Pole Circle or NPC. An identical part S, created by the dividing plane going through the South Pole Circle. the central part C, the largest part, which includes the equator Two more parts are considered: A large part NC, which is the total of both N and C. A large part NS, which is the total of both N and S.

1. Let the Earth be cut by a plane NPC into two parts N and CS, and their weights one towards the other will be mutual equal.
   The most difficult part of this sentence is the definition of the word weight. The problem is the mass (volume) of each part is different but the weight is the same.

   For the other plane SPC, parallel to the former NPC, the greater part CS is cut off into two parts C and S, whereof S is equal to N, first cut off, it is evident that the middle part C will have no propension by its proper weight towards either side, but will hang as it were and rest in an equilibrium between both.

   In this sentence the difficult part is the word propension (natural inclination, Webster) and the word proper weight. The tricky part is why this specific division? Why are the two outmost parts N and S equal?

   But the one extreme part S will with its own weight bear upon and press the middle part c towards the other extreme part N; and therefore the force with which SC, the sum of the parts C and S tends towards the third part N, is equal to the weight of the part S, that is to the weight of the third part N.

   Again also here: Why are the two parts N and S equal?

   And therefore the weights of the two parts SC and S, one towards the other, are equal, as I was to prove.

   That means the mutual attracting forces are equal. But what exactly did we physical prove?

   And indeed if those weights were not equal, the whole Earth floating in the non-resisting ether would give way to the greater weight, and retiring from it, would be carried off in infinitum

   See also: Reflection 3 - The third Law of Newton.

10. Additional amplifications - page 35

10.(a) The proportionality of mass and weight and the experiments on the pendulums - page 35

Proposition XXIV. Theorem XIX - page 35

The quantities of matter in pendulous bodies, whose centres of oscillation are equally distant from the centre of suspension, are in a ratio compounded of ratio of the weights and the squared ratio of the times of the oscillations in a vacuum.

Okay.

page 36

On the other hand the force acting vertically downward on the pendulum of length l, displaced by a distance a from the vertical, is given by

downward force = Wg

Where W denotes the gravitational mass and g the value of gravity

What the whole text indicates that the movement (# of ticks over a fixed period) of pendulums, placed on the surface of the earth, assuming there is no energy loss is constant and a function of the attracting force between the pendulum and the earth.

page 37

The experiments themselves are described in greater detail to Section VI book II and more briefly in Proposition VI, Book III. This continued and repeated reference to his experiments on the pendulums shows the importance that Newton (justly) attached to them.

What this means how important experiments are.

10.(b) Maxwell's reformulation of Newton's Laws of Motion - page 37

10.(c) The Newtonian principle of relativity - page 41

One can, on this account, say that inertial frames are undistinquished: any one frame will serve as equally as any other

My conclusion is that to understand the behaviour of moving objects, only use one frame.

18. The dynamics of a particle under a general law of centripetal attraction - page 58

We shall assume that the force acting on a particle of inertial mass, m, at a distance, r, from the centre of attraction is:

The importance is the particle has an inertial mass, m

32. The mapping of orbits described in the complex plane - page 11

96. Rules of reasoning in philosophy - page 345

97. Phenomena - page 346

page 347

This paragraph describes 6 phenomena.

Phenomenon III That the five primary planets, Mercury, Venus, Mars, Jupiter, Saturn, with their several orbits, encompass the Sun.

Newton describes here that the Sun is the center of the Solar System.

Phenomenon IV That the fixed stars being at rest, the periodic times of the five primary planets and (whether of the Sun about the Earth, or) of the Earth about the Sun, are as the 3/2th power of the mean distances from the Sun.

The fact that Newton mentions 'fixed stars' implies that his world view is larger than only our solar system.

103. Propositions VI: The confirmation of the equality of the inertial and gravitational masses by astronomical data - page 362

page 362

Proposition VI. Theorem VI That all bodies gravitate towards every: and that the weights of bodies towards any one planet at equal distances from the centre of the planet, are proportional to the quantities of matter which they severelly contain

It was, as we have seen, his first concern in formulating his laws; and towards which he first carried out the first precision measurements with the pendulum. (described in detail in the General Scholium at the end of Section VI of book II)

The folowing text is important because the mass of a pendulum is considered gravitational mass.

104. Propositions VII: The universal law of gravitation - page 370

Proposition VII. Theorem VII That there is a power of gravity pertaining to all bodies, proportional to the several quantities of matter which they contain.

106. Propositions X-XIV - page 375

page 376

Hypothesis I That the center of the system of the world is immovable

In a slightly different wording: That the center of the universe is fixed or 'absolute'

Proposition XI. Theorem XI That the common center of gravity of the Earth, the Sun and all the planets is immovable

Also here the same comment: is fixed or 'absolute'

page 377

If the 'system of the world' (as Newton once stated) consists of the 'whole space of planetary phenomenon' then the frame in which the centre of gravity of the system is at rest is clearly the most useful as Newton argues in Proposition LXVII (in context of the three-body problem) and in corollary of Proposition LXVIII (in the context of many bodies like Jupiter and its satellites).
We adopt the same principle in considering larger systems: the motions in a star cluster with respect to its center of gravity; the motions in a galaxy with respect to its centre; and ad infinitum

From Newton's point of view the center of gravity of the solar system is at rest. You can also call it fixed or 'absolute'.

Reflection 1 - The book GRAVITATION by MTW

Parts of the book "Newton's Principia for the common Reader" is also discussed in the book review of the book GRAVITATION by MTW. See: BookReview: GRAVITATION by MTW
Specific:

• $1.3 Weightlessness - part2
• $1.5 Time - page 26
• Reflection part 1: Final page
• Reflection part 2: Who is right Newton or Einstein or both

The book: "Newton's Principia for the common Reader" is also mentioned in this Article Review: Article_Review_Sparks_of_Artificial_General_Intelligence--Early_experiments_with_GPT-4
Specific at this link: Reflection 2 - Understanding and Intelligence

Reflection 2 - absolute versus relative in Newton's opinion

IMO Newton' makes a clear difference between absolute versus relative:
The behaviour of a large set of objects is absolute. The behaviour of a small set of objects, part of this large set is relative.
As such:

• The behaviour of the universe is absolute and at rest. The behaviour of the stars of in our Galaxy is relative.
• The behaviour of our Galaxy is absolute and at rest. The behaviour of the Sun is relative.
• The behaviour of the Sun is absolute and at rest. The behaviour of the planets is relative.

In Einstein's view all of the above is relative, mainly based on the view, observations or experiences of the many observers. Space-time is absolute (i.e. specific the geometry of space-time (?))
Einstein calls matter or mass an invariant or an absolute. Newton would agree with him.
The problem is to calculate the mass based on observations for each theory.

Reflection 3 - The third Law of Newton.

The third Law of Newton is discussed in:

• 8.3 Law 3 - page 23
• 9. The scholium to the Laws of motion - page 30

The third Law writes: To every action there is always opposed an equal reaction.
What that means is that both action and reaction are always happening simultaneously.
The main problem with any law which describes something that is always the case or is true, that such a law does not explain anything. What is even more, such a law can not be used as an argument (pro or contra) to explain any other scientific experiment.

When we consider an object on the surface of the earth and you want to pick it up, you feel a force attracting that object to the center of the Earth. At the same time there exists also a force attracting the earth towards the center of the rock. Is that force what is meant with action and reaction in Newton's third Law?

The third Law of Newton is specific discussed at Page 32 . There is mentioned that a part of the proof is based on experience. Experience is a rather loose way to do science.
At that same page also a demonstration is discussed considering three objects: A, B and obstacle. The problem is when two objects are considered in relation to gravitation and a third object is involved, than that third object will also influence the two other objects. This makes any conclusion related to Newton's third Law tricky, if not impossible.
At Page 33 the same problem pops up. Also here three objects are discussed. This makes any discussion, where equilibrium is involved, tricky.
The problem is when you cut the earth in two, along a flat plane, nothing physical will happen. You can assume that both parts will attract each other with the same force, but you can also assume that there is no force. The point is that it is difficult to demonstrate that gravitation is mutual. Two experiments maybe can help:

• Lift a stone up from the surface and claim that the attracting force from by the earth towards the stone and from the stone towards the earth are identical and in opposite directions.
• Bring both parts in circulation around each other (binary system) and based on observations calculate that the attracting forces are identical.

What I want to point out is that there is nothing wrong with the third law, but to demonstrate specific when this laws applies is tricky.

Reflection 4 - The book GRAVITATION by MTW

Parts of the book "Newton's Principia for the common Reader" is also discussed in the book review of the book GRAVITATION by MTW. See: BookReview: GRAVITATION by MTW
Specific:

• $1.3 Weightlessness - part2
• $1.5 Time - page 26
• Reflection part 1: Final page
• Reflection part 2: Who is right Newton or Einstein or both

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