The next important problem I have solved is another one made famous by Newton, although this time he invented it without much help from the Greeks. By analyzing a diminishing differential applied to the arc of a circle, Newton claimed to prove that as the arc length approached zero, the arc, the chord, and the tangent all approached equality. I have shown that Newton's analysis is false. Newton monitored the wrong angle in the triangle created, which skewed his analysis. He did not notice that another angle in the triangle went to its limit before his angle, assuring that the tangent remained longer than the arc and chord all the way to the limit. This solves, all at once, many of the mysteries of trigonometry. Newton's ultimate interval, which became the infinitesimal and then the limit, is proved by me to be a real interval, where the variables do not go to zero and they do not go to equality. This is the reason we find real values for them. Even at the limit, the tangent is not zero and it is not equal to the arc or chord. The tangent and the arc are expressed by two different (perhaps infinite) series of differentials, and these series do not approach zero in the same way. In fact, one reaches zero after the other one, which makes it a lot easier to understand why the equations work like they do.
Because Newton misunderstood circular motion in this way, he also misunderstood the dynamics of circular motion itself, and the equation that expressed it. His basic equation a = v2/r, which is still the bedrock of circular motion, is wrong. If you express the orbital velocity as v = 2πr/t, then the equation must be correct, of course. We know that from millions of experiments. The problem concerns the fact that that variable cannot be a velocity. A velocity cannot curve. The circumference of a circle cannot be expressed by a simple velocity, even though the apparent dimensions of the variable (m/s) would imply that it could. Velocity is a vector, and there is no such thing, mathematically or physically, as a curved vector. By definition, a velocity can have only one spatial dimension. Any curve must have two spatial dimensions. Of course a velocity has a time in the denominator, which gives it two total dimensions. A circumference or orbit must have at least three dimensions (x,y,t).
Flying in the face of this very simple fact, for some reason Newton assigned 2πr/t to his velocity. To add to this error, he conflated the tangential velocity with the orbital velocity. Going into the series of equations that proved a = v2/r, he defined v as the tangential velocity. That is, it was the velocity in a straight line, a vector with its tail touching the circle at a 90o angle to the radius. But at the end, he assigned v to the orbital velocity, which curved. Any elementary analysis must show that the orbital velocity is a compound made up of the tangential velocity and the centripetal acceleration. In fact, Newton said so himself. It is a fact we still accept to this day, and it is taught in every high school physics class. If so, it cannot be the tangential velocity and it should not be labeled v.
This is of paramount importance for any number of reasons, but I will mention only a couple. Since contemporary physics has inherited this confusion of Newton and utterly failed to correct it or notice it, all our circular fields are compromised. I have shown that Bohr's analysis of the electron orbit is affected by this mis-labelling, and that the equations used to calculate the velocity of quanta emitted by electrons must be falsified. Huge problems have also been caused by the ubiquitous equation ma = mv2/r. The form of that equation has led many to think that the numerator on the right side is a sort of kinetic energy, but the mv2 comes from Newton's equation, and the velocity is not really a velocity. It is not a linear velocity, but it is also not an orbital velocity. It is simply a mis-defined variable. It is not a velocity of any kind. It should be labeled as an acceleration. By correcting Newton's proof, I discovered that
vt2 = a2 + 2ar
ao2 = 2acr
ac = ao2/2r
Where ao is the orbital acceleration, replacing the misnamed orbital velocity, and ac is the centripetal acceleration.
By cleaning up our variables and definitions, we can avoid many problems. Just as a starter, the equation ma = mv2/r must become ma = mao2/r. That keeps us from thinking about kinetic energy when we look at the right side, and solves many many errors, including several of Bohr, Schrodinger and Feynman.
And of Euclid:
These findings are both more fundamental and more inventive. To add yet another level of tidiness, I will begin with the oldest problem I have solved: meaning the problem that had persisted for the most amount of time before I solved it.
That oldest mistake is one that Euclid made. It concerns the definition of the point. Entire library shelves have been filled commenting on Euclid's definitions, but neither he nor anyone since has appeared to notice the gaping hole in that definition. Euclid declined to inform us whether his point was a real point or a diagrammed point. Most will say that it is a geometric point, and that a geometric point is either both real and diagrammed or it is neither. But all the arguments in that line have been philosophical misdirection. The problem that has to be solved mathematically concerns the dimensions created by the definition. That is, Euclid's hole is not a philosophical or metaphysical one, it is a mechanical and mathematical one. Geometry is mathematics, and mathematics concerns numbers. So the operational question is, can you assign a number to a point, and if you do, what mathematical outcome must there be to that assignment? I have exhaustively shown that you cannot assign a counting number to a real point. A real point is dimensionless; it therefore has no extension in any direction. You can apply an ordinal number to it, but you cannot assign a cardinal number to it. Since mathematics and physics concern cardinal or counting numbers, the point cannot enter their equations.
This is of fundamental contemporary importance, since it means that the point cannot enter calculus equations. It also cannot exit calculus equations. Meaning that you cannot find points as the solutions to any differential or integral problems. There is simply no such thing as a solution at an instant or a point, including a solution that claims to be a velocity, a time, a distance, or an acceleration. Whenever mathematics is applied to physics, the point is not a possible solution or a possible question or axiom. It is not part of the math.
Now, it is true that diagrammed points may be used in mathematics and physics. You can easily assign a number to a diagrammed point. Descartes gave us a very useful graph to use when diagramming them. But these diagrammed points are not physical points and cannot stand for physical points. A physical point has no dimensions, by definition. A diagrammed point must have at least one dimension. In a Cartesian graph, a diagrammed point has two dimensions: it has an x-dimension and a y-dimension. What people have not remembered is that if you enter a series of equations with a certain number of dimensions, you must exit that series of equations with the same number of dimensions. If you assign a variable to a parameter, then that variable must have at least one dimension. It must have at least one dimension because you intend to assign a number to it. That is what a variable is—a potential number. This means that all your variables and all your solutions must have at least one dimension at all times. If they didn't, you couldn't assign numbers to them.
This critical finding of mine has thousands of implications in physics, but I will mention only a couple. It has huge implications in Quantum Electro-Dynamics, since the entire problem of renormalization is caused by this hole in Euclid's definition. Because neither Descartes nor Newton nor Schrodinger nor Feynman saw this hole for what it was, QED has inherited the entire false foundation of the calculus. Many of the problems of QED, including all the problems of renormalization, come about from infinities and zeroes appearing in equations in strange ways. All these problems are caused by mis-defining variables. The variables in QED start acting strangely when they have one or more dimensions, but the scientists mistakenly assign them zero dimensions. In short, the scientists and mathematicians have insisted on inserting physical points into their equations, and these equations are rebelling. Mathematical equations of all kinds cannot absorb physical points. They can express intervals only. The calculus is at root a differential calculus, and zero is not a differential. The reason for all of this is not mystical or esoteric; it is simply the one I have stated above—you cannot assign a number to a point. It is logical and definitional.
This finding is not only useful in physics, it is useful to calculus itself, since it has allowed me to show that modern derivatives are often wrong. I have shown that the derivatives of ln(x) and 1/x are wrong, for instance. I have also shown that many problems are solved incorrectly with calculus, including very simple problems of acceleration.
I would of course recommend spensing time with Miles' Work on his website. You may not agree with everything he writes, but he has just that edginess to provoke you to do your own independent thinking.
We all suffer from the legacy we inherit, particularly in cultural and the more insinuous language terms. Much of what Miles attempts to clarify result from the subtleties of language, translations, redactions , opinions and apprehensions. It requires a zeal to sift through all of this , establish working hypotheses and then to correct accordingly. In addition to update and correct on the basis of new empirical data, new insight and new apprehension. One has to be driven o do this, otherwise it just won't get done. Miles is the man who is driven to do this. His is a work in progress with little reward or recognition in official science media. He walks a lonely road with a few compadres.
Does he get it right? I am afraid i must answer a philosophical question philosophically: Judge for yourself!
Euclid as i have discussed, did not write a mathematical treatise but an astrological one, he did not write to collect just pragmatic methods into a useful manual, he wrote a university level course in Platonic Philosophy from the doubtful theory of Ideas or Form both Socrates and Plato accepted philosophically. The Stoikeioon makes the theory abundantly clear in its ramifications.
Newton was a Student principally of Euclid and thus of Plato, and he never claims to have all the answers, nor to have it all correct, especially as he wrote in haste what he had been meditating on for years, questing and questioning. The fundamental importance of the spiral and the other conic section forms and surfaces and spaces,especially the sphere and the circle was appreciated by Newton. He says many surprising things about them in his writings, all of which are subtle, both in the latin and the English translations. It is this subtlety that others have struggled with. Unfortunately Newton's subtle mind "went" as he got older, and certain details and connections he clearly forgot. For this reason, Cotes and De Moivre need to be consulted to get a fairer idea of what he meant or any possible and likely mistakes he made.
Asserting something to be so does not i am afraid make it so. Thus when we propose a thing we ought to demonstrate it. But to demonstrate an insight we must communicate the parts of that insight, and this we do with definition , demands, agreed ways of judging, lemmae( ie just chew on this for a while) conans , axles (of axioms) etc.. In the process it may occur that one agrees with the result proposed, but this could range from precise agreement to broad agreement. It may also be that a step in the proposed demonstration lacks empirical evidence or subjective certainty, and one therefore flags this up, the outcome of doing this always being uncertain! This is a communication process. When does it become a teaching process? When the power relationships change to impose this. Thus the use of pedagogy, coercive behaviours and language, to enforce a view or system of thought is not Philosophy. Neither is the use of sophistry or hypnogogic practices to persuade. Language is replete with these techniques and technologies and so is so called "Logic". The notion of "The Truth" is also not free from this coercive force.
Mr. Cotes, acquainted not with just part but all of Mr. Newtons works writes a cogent preface in which he sets out the gravitational force as what we now call a field, and that field has Fractal Structure and cause and effect assignments to the gravitational force due to its mutual nature between bulks! That one may freely roam between the centripetal (effective) force on a body in motion about another and the attracyive (centrifugal and causative) force of a body around which others move in orbits, and that we may ascribe the same magnitude of force to each notion but operating in opposite direction mentally(subjectively), and to that we must always bear in mind the mutuality of the circumstance which is best represented by some middle space that attracts both bodies in rotation around it, ought to signify to us the fractal subtlety of the circumstance in which we come to be and which we attempt tp interpret both objectively and subjectively.
For this reason Mr. Newton thought it best to publish in 3 distinct volumes his principles, his methods and hiw Metaphysical interpretations(The systems of the Worlds). Mr Cotes, therefore, being apprised of all three works sets out admirably and concisely his later view of the principles after so long a time of foment and controversy , which all seem to agree has vindicated Mr. Newton and exonerated him of all subterfuge. This is not to say that he has it just right, for not eve Mr. Newton makes such a claim; looking towards his successors to more perfect or indeed better philosophize than he. It is therefore just that in his atter years he should be so esteemed, not for the correctness of his theory, but for the humility of his approach to all, saving of course that he is no saint!.
It is also to be noted that Mr . Newton Prefaces his work with comments on the ancients and the moderns and refers to the Geometry and its less perfect relative Mechanics as being one a gateway to the help of the other, that what is done by an artist or artisan, being mechanical, nay be guided toward perfection, by due diligence, by the principles of Geometry. Thus, in other words, the practical and real may taske wise direction from the formal and immaterial(not real in that sense of the other, but subjectively real). From these 2 a better philosophy may be constructed, which Mr Cotes believes Mr Newton has admirably achieved.
Mr. Cotes and i do not agree on all interpretations of Mr. Newtons philosophy, for he being a straight Yorkshireman appreciates that which is straight and not occult and i being inclined otherwise apprehend hat which is spiral and not occult. Thus when language allows, such as in the use of the term "right", i am allowed to be curvilinear , where Mr Cotes would be "straight". I accept that this is the opinion also of Mr. Newton, but only in the application of tangents, not needfully or necessarily in the innate natures of "Quantities of Motion", that is the quanta of moion not the measure of it.
Newton himself hopes for an explanation of the "stickiness" of particulate matter, and therefore a better philosophy. for he is not unaware that attracting forces do not explain the "stickiness" of particles bodies by bulk, for it is obvious, if considered, that a mote of dust cannot attract another mote o dustby very much gravitational force. Secondly, what is to prevent attracting bodies from continuing to attract and compress one another into miniscule space ad infinitum? Thirdly. what is the clear and admissible link between attractive force and pressure, so that a centripetal force is pressure effect on a body caused by an attractive force by another body and vice versa or the space in between them?
The notion of Force is in fact different to pressure, the first being an innate nature of motin, the second an external action of motion transferred from another colliding body. What unites them is the resultant acceleration of the bodies in question. Thus in every way, to pursue what is set before us analytically and synthetically we must establish definitions etc, and the principle ones are velocity and acceleration. However, such a definition must account for all the different distinguishable motions of a line, and to this end Mr Newton applies himself to ascertain how his might be achieved in a relational or relative way, so that one may transform from one sor to any other sort of motion SAFELY . Hence the lemmae, and the method of fluxions, which he more nearly explains elsewhere.
Read, dear reader and philosophize. Make your own sense of your experiential continuum.