Newton's Scholium

The 'last Scholium' (Often Ignored)

from 'Section 12 - Spherical Bodies'

of 'Book 1 of Newton's Principia'.

This section results from a theory by Feng and Gallo, which appears NOT require exotic new mass, such as 'Dark Matter'. By using Newton's advice, as written more than 200  years ago, it achieves results that  replace:

some or all of the erroneous concepts of 'Dark Matter',


the undetectable particles such as 'WIMPS'.

1.1 Newtons "Principia"

In 'Book 1', Newton stressed that the rules given in

       "Section 12, - The Attractive Forces of Spherical Bodies" are for Spherical bodies only;

(note: 'point particles' are also spherical).

        Section 12 introduces the formula for the gravitational Force between two spherical bodies :-

                                                                F = m1 x m2 x G


       where   'F' is the force between two spherical particles, which have masses 'm1' and 'm2',

                     'G' is the 'Gravitational Constant' and

                     'r' is the distance between 'm1' and 'm2'.


Newton described the constraints associated with such spherical bodies:

If either these bodies are an infinite distance away, they can be described as being a point.

This also is a good aproximation when they are much closer.


It really is only when they are at distances that is not large compared to the actual spacing between them, that we have to think of them as 'spherical' shaped objects.

- If two spherical bodies are NOT in contact with one another, then they can be treated as two point objects.

-If one sphere lies entirely inside the other,  then the inner one can be considered to be a point. However, the forces between them are not so obvious:

If the centre of the  inner sphere lies at a distance 'd' from the centre of the outer sphere, then a very specific and fortunate equality arises:

the forces from every point on that inner sphere when integrated over the whole body equals zero.


Newton included a ‘Scholium’ (foot note), at the end of ‘Section 12’,

which warns that the equations in that section are for spherically-symmetrical bodies only.

If either of the bodies (m1 or m2) is 'NOT spherically symmetric',

then it is necessary to modify the formulae to suit the non-spherical symmetry. 

It certainly does NOT say that 'Physicists should ignore  this advice';

as is a very common  practice in Modern Physics.

The failure to use this scholium  may be one of the major causes of the need to introduce 'Dark Matter'.

Before discussing any of the possible properties that may account for Dark Matter, let us look the variations various authors have used when discussing the above equation.

1.1    Newton

Newton: Warned that the equation is ONLY appropriate for spherical bodies.

1.2    Vera Rubin

'Dark Matter' was first discussed in the 1930s and only occasionally mentioned through to 1968, when an astronomer, Vera Rubin, resurrected it. Vera was an experimentalist and did a lot of excellent work measuring the impact of gravity on the velocity of stars within galaxies.

In a short autobiography, published by 'Physics Today', she confirmed her approach to the gravitational forces on stars:

“Isaac Newton showed that the force on a mass, at radius 'r' from the centre of a symmetrical mass distribution, is proportional to the mass inside of that radius 'r'.”

But note, this applies to masses with spherical symmetry only.

However, Rubin didn’t distinguish the difference between spherical and 'disk-shaped' galaxies.

She says

“High school students learn that, in a gravitationally bound system like our solar system, a planet, moves in a closed orbit such that M/G=rV^2 where 'M' is the mass of  the sun, 'G' is the gravitational constant and 'V' and 'r' are the velocity of a planet and its distance from the centre of its sun.”

In a subsequent paragraph she says that, this is because the symmetry of a sun is more or less spherical.

That is all correct!

However, she then continues;

“In M31, the same relation between mass velocity, and distance holds.”

M31 is a galaxy; that is  more 'disk-like' than 'spherically symmetric'.

​That is wrong!

Solar bodies are rotating around a spherical body (a sun);

Galactic bodies are rotating around a shape that is usually described as 'disc-shaped'.

She repeats this erroneous comparison, either explicitly or implicitly, in many of her other papers, such as: [2] and [3].

Vera’s erroneous use of ‘Dark Matter’ has been consistantly used without question, by a large number of the modern Physicists. A huge percentage of physicists are 'un-knowingly' making the same mistake that she made.


To correctly use Newton’s theory on a disc , she should have extended the caculation, as implied by Newton’s Scholium, so that it supports a disc-like galaxy and not just spherically symmetric bodies.

1.3 Feng and Gallo

Feng and Gallo, used Newton’s Scholium more or less as advised [5]. They treated each position in the galaxy as a ‘point particle’. Then the force between each body and the 'observer', can be calculated correctly, using Newton’s spherically symmetric formula. They more or less 'integrated' the forces and their calculations no longer seemed to required Dark Matter.


However, there are problems:

1. The disks they use are like a round coin; there is a flat 'verticle' surface right around the circumference, which is unlike any galaxy. They realised that this will perturb the calculation. So they specifically examined various shapes at the boundry.  These problems were judged to be NOT significant in the examples they considered. [??]

2. They also realised that a lot features associated with galaxies are not covered by there model:

- Stars are not continuous within the galaxy; they are discrete objects within the galaxy.

- Their galaxy models didn't have spirals, bars, a bulge at the centre etc, as seen in real galaxies.

- New stars are created at the edges of a galaxy and old stars die at the middle, creating a continuous migration from the outside towards the centre.

3. Their method of calculating the total force on an observer assumed a continuous density of matter and not a scatter of disctrete stars (solar systems). This allowed them to replace the summation of billions of stars that exist in galacies, by a simple integration in which they can try various densities etc. 


1.2.       A Pedagogical Model

This is a simple model that can be used to illustrate the mistake that Rubin has made (and others are still making).

Firstly, place three stars, 'S', in a 'linear' structure and 'integrate' (sum) their forces. For simplicity, we shall assume, that the product (m1 x m2 x G) equals 1', so that each of the forces of interest reduce to

F ≈ I/r^2.

Firstly assume that the three stars are equally placed in a row at positions (-1), (0) and (+1), so that this 'galaxy' is 'linearly symetric' NOT 'spherically symmetric'.

Place an observer ‘O’ at a position outside of but in the same line as the stars; at (+2).  

                    S              S            S           O

                   (-1)          (0)         (+1)      (+2)     

Then calculate the force (F) experienced by the Observer for these three stars:

                   F (-1) = 1/(3^2) = 1/9 = 0.11,          

                   F (0)  = 1/(2^2) = 1/4 = 0.25 ,          

                   F (+1)= 1/(1^2) = 1/1 = 1.0.           The total Force = 1.36.

Alternatively, place all three stars at F(0) which is the ‘Centre of mass’. This creates a single body with the same mass, but it is totally spherically symmetric. 

                   -              SSS            -              O                                            

                 (-1)            (0)            (+1)          (+2)

The forces are now:

                 F (-1) = 0*1/(3^2) = 0,           

                 F (0)  = 3*1/(2^2) = 3/4 = 0.75 ,     (This is now effectively a 'single particle')

                 F(+1) = 0*1/(1^2) = 0.                     The total Forcee = 0.75.  

It is significantly smaller, (~54%) .

Even in this extremely simple model, the calculation, which follows Newton’s advice for non-spherical bodies, gives a much larger force (F = 1.36) than the one (F = 0.75) where all of the stars are assumed to be based at a single point.


This simple model shows that Newton’s formula, F = (m1 x m2 x G) / r^2 only applies to spherically symmetric bodies. The non-spherical bodies produce bigger than expected forces, a difference that, over the last few decades, has been wrongly attributed to 'Dark Matter'.



Looking at the calculations more closely, we can see that this difference occurs because

  • the right-hand particle, (+1) is very much closer to the observer than the other two stars. Because of the non-linearity of the 1/r^2 term, this 'close' star returns a dis-proportiionately large force.

  • also the left-hand star produces a dis-proportionately smaller contribution,

  • The central star gives an intermediate contribution, which is not  the average of the previous two.


The 'closer' stars always dominate the 'more remote' stars. This larger total force has been wrongly assigned to 'Dark Matter'.

This was kind of problem that  Sir Issac Newton anticipated in Book 1 of his Principia,

(in the last Scholium in 'Section 12, ‘The Attractive Forces for Spherical Bodies’) .



1.3      Real Structures

Rubin and many others expected the 1/r^2 term to work for non-spherical bodies, because they worked so well for other highly symmetric structures such as planetry orbits, where the sun, planets and moons are (approximately) spherical bodies and mostly lie in the same plane.


However, when they came to work on galaxies, which are more disc-like than spherically symmetric, the predictions made without considering Newton's scholium, were significantly smaller than their research measurements.

The above ‘Pedagogical Models’ shows a difference of a factor of almost x2. 

If our 1D model is replaced by a 2D disk, we would expect a factor of ~x5.

More comprehensive theoretical models, such as Feng and Gallo's, show more realistic ratios of between x5 and x10.


1.4.      Feng and Gallo's Calculations.


Feng and Gallo [4] have respected the Scolium and shown how to estimate the forces,  in a  'disc-shaped' galaxy . This is the first of two series of papers:

- Papers, where the authors are listed as 'Feng & Gallo', are strongly technical whereas

- Papers, where the authors are listed as 'Gallo & Feng', are reviews, mostly for conferences etc.

The most significant contribution that Feng and Gallow made, was to integrate the forces across a 'disc shaped' galaxy, rather than using the total 'mass' at some central point, which had already been shown to be incorrect. In successive papers, their use of the technology becomes stronger and less approximate. Their results have been  applied to a number of diferent types of galaxies [5].

The integration of the 1/r^2 term is needed in systems where the attracting mass is evenly distributed but does not lie in a spherical distribution. However, it does not model the following features:

spiral structures,        bars,         the inner bulge    or      black holes.    

These commonly occur in real situations, but are not usually as significant as the need to integrate  the local forces, which adds the most fundamental contribution from Newton's Scholium.


One of the most popular alternatives to Feng and Gallows theory is attributed to unknown cosmological bodies known as 'WIMPS'. These are said to form a 'halo' in the plane of the disc  and outside the visible structure of the galaxy.


No one knows what a WIMP is. It appears to be another piece of QM magic. This is a significant problem.

One feature that has been used to favour WIMPs is the refraction of light as it passes close to the sun. Einstein predicted it and experiments have confiirmed that light is sensitive to local gravitation. Light from another galaxy was bent when it passed close to our sun during an eclipse of the sun.

It is NOT a perogative of WIMPs, but just an attribute  that any model could add in.

It will also apply to the theory I have used in this paper.

It is a universal property, that a photon will be attracted by the gravity of any large astrononmical body  if it passes just a short distance away from it.

1.6 Which is the most valuable model?


Vera Rubins explaind the galactic motion as though it ie entirely spherical.

Feng and Gallo can explain the approximate nature of the force involved within a galaxy, but they treat the galaxy as a continuous body.

The WIMP team hypothesise with almost no mathematical functions describing the unknown particles as though they have relationships.

The difference lies in how the combined effect of all the elements of a body interact with one another.

Looking at the integrations involved above, one sees good reasons why they give different answers.

1.6.1 Two infinitely small particles

A recent review in New Scientist ​​[6], asks the question "Does Dark Matter Really Exist?".

The articles replies:

"By far the bigger fraction (of cosmologists) thinks the discrepancy is down to the influence of a mysterious gravitating 'Dark Matter' that we have yet to observed directly."

It continues

"But a rebelious minority believe that dark matter is an illusion and that galaxies maintain their shapes because of a new facit of gravity we have yet to understand."

​This current article supports the 'rebels' who believe that Newton, when recommending his Scholium, understood the need for this extended 'facit of gravity'.


Criticisms of the F-G proposal include:

1. The 'thin-disk' galactic model is not representative of spiral galaxies in general.

'Thin Discs', are a lot more representitive than these physicists realise. F-G papers confirm this.


2. External Hydrogen is often writen off as 'Dust' and ignored by many researchers, Rubins included.

F&G have studied this in their model, but finally dropped the idea for a later study.


3. The Mass/Luminocity ratio drives many of the models; it is a crude approximation.

F&G use it carefully and have written at least one paper showing now reliable it actually is.


4. Modern spiral galaxy models are not necessarily restricted to Keplerian orbital assumptions.

This is the most significant feature of F&G's theory.

The orbits should NOT be restricted to Keplarian rules,

but should expand them using the Scholium in Section 12.

5. A large proportion of the Physics Courses on the web, restrict themselves to the Keparian Rules.

Vera Putin has publically rejected variations from the Keplarian  rules.

6. Some of the predictions differ significantly from empirical data.

This is expected from most theories , particularly in their early usage.  

However, such deviations are far closer to the mark than 'WIMPS' for example.

7. Gravitational Lensing:

This is NOT specific to DM???????? It all starts with

It explains the missing gravitational forces needed to explain the measurements of force.

Following Newton's advice about his Scholium also explains this without anything dark.

It explains the additional gravitational force involved in 'Lensing'.

In either theory, Lensing is just some distortion resulting from the same  structures.

It explains the additional forces from the External Hydrogen.

This also results from the non-visible Hydrogen not involved in any fusion reaction.


1.6.      Dark Matter Conclusions


​​As one of the 'rebels', the author of this paper thinks that 'Dark Matter' is probably not required.

It is partly the modern Physicists, who are contributing to the misunderstanding, by overlooking that critical 'Scholium' in Newton's 'Principia': 


This explains the additional forces ('Dark Matter'),

It works 'hand in hand' with the 'External Hydrogen', (pointed out to Vera Rubin by Mort Roberts)  associated with the outer reaches of galaxies, where a huge ring of hydrogen is just ignored.



3     References.


[1]  Rubin, V.                                   Physics Today, p8, December, (2006),


[2]  Rubin V. and W. K. Ford,        Astrophys, p159, 379,  (1970).

[3]  Rubin V., Burstein, D. Ford, Jr. and W. K.; Thonnard, N.

       "Rotation Velocities of 16  SA Galaxies and a Comparison of Sa, Sb & Sc Rotation Properties".

                                                                 The Astrophysical Journal. 289: 81ff. (1985).


[4] Roberts, M. S, and Whitehouse, R.N.,  

     Astrophysical Journal, Vol. 201, p. 327 - 346  (1975)

[5] James Q. Feng, C. F. Gallo


"Modeling the Newtonian Dynamics for Rotation Curve Analysis of Thin-Disk Galaxies"

     arXiv 11 August, 2011;

[6]  James Q. Feng, C. F. Gallo            

"Galactic Rotation Described by a Thin-Disk. Gravitational Model without Dark Matter"  

                                                         arXiv  8 Oct (2014)

[7]  James Q. Feng, C. F. Gallo     

        "Mass distribution in rotating thin-disk galaxies according to Newtonian dynamics"

   Galaxies 2, 199-222 (2014), arXiv (Jun 2014)


[8] James Q. Feng, C. F. Gallo

"Deficient Reasoning for Dark Matter in Galaxies"

     arXiv Oct 2014


[9] Clark, Stewart                                                   

       "Does Dark Matter Really Exist?"

    New Scientist (2018) p33.


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