The gyroscope and the coupling of forces and momentum, angular or otherwise.
One of the mistaken notions that students of Newtons ideas came up with was that points travel in straight lines by default. These are point masses or rather centres of symmetry. However, Newton did not seek to imply this inanity. The first axiom is just that: an axiom by which one may begin to apprehend the full complexity of the inertial forces of the Aether.
To say also that Newton did not accept the Aether, and to site his dispute ith Huygens is also another great misrepresentation!. The differences, and their were many, we're philosophical, and methodological. The proposition of waves without empirical data was a bold guess. Newton could make bold guesses till the cows came home! What he could not do was empirically substantiate all his guesses. Huygens gave a complete argument for his theory which relied on drawings and rational/ divine reasoning, not on empirical measures of quantity. The question was: how are these waves to be measured? To this Huygens gave no convincing argument, none that could not be equally demonstrated from the corpuscular reasoning. Corpuscles could be viewed under a Hookes Micrscope at the time. Luminiferous aether waves could not.
While the lens fringes did cause a difficulty in explanation, when quantified , because they did not agree with mathematical prediction, in those days, mathematics was hardly a baby in the cradle! It had no real authority to prove or disprove a theoretical position. As such it was therefore not an issue that one set of equations gave a better approximation than another. What was deemed of greater importance by Newton, was the empirical data, and the soundness of the method of reasoning. Appeals to occult or divine laws were not acceptable. Divine laws had to be manifest in nature not in Theology!
Newton's confusion at not being able to demonstrate fluid motives in the rarefied conditions of space, as he assumed , lead hom to wonder how forces could act at a distance in the aether. In this he assumes aether to be non physical, and thus akin to empty space. Later scientists developed the notion of a physical aether, but they could not detect the assumed results, or so they say.
Apparently Goethe and Steiner had an alternative theory to matter which did not give these particular difficulties, but created others. The gyroscope is at the heart of a fundamental revision of Newtonian fluid motion, which Newton left for others to resolve.
The one thing that Newton's naive students failed to recognise was his vector analytical tools. About the only scholar who did deeply imbibe them ,apart from De Moivre and Cotes, was Lagrange.. The consequence of this was that 2 naive students of Lagrange, both develop what we now call the linear Algrbra. Both develop the concepts differently nd in different degrees of generality, but the motivation was the same. Grassmann sought to justify his Fathers work( but ended up showing up his errors! ) while Hamilton sought to justify his friends work on imaginary logarithms.
Of the 2 Grassmann was the more philosophically thorough. While Hamilton sought to be mathematical in his rhetoric, he did do a version which was more " wordy" for more " public consumption". This he called mathesis, a rare word even in the Greek. This was his doctrine of Algebra. He created the subject of Algebra, as opposed to the style of symbolic representation of arithmetic, hitherto called Algebra.
Grassmann was of the old school. Algebra meant symbolic arithmetic to him and his brother, and indeed all Prusdian intellectuals. Thus the De Morgan's, the Hamilton's of the British empire were forging a new kind of Algebra, typified by Boole, who used a rhetorical or grammatical algebra to develop Boolean Algrbra, and the laws of thought.
However the focus of all three was the same: to explore the contribution of subjective thinking to the development of systems usually called mathematical.
Mathematicians site this as a golden era, principally because it saw the advancement of the status of the subject, and thus their reputation and influence. Mathematicians tended to be rare or solitary , but mostly highly regarded in many fields. This was because mathematics hold not have divorced itself from astrological computations. By doing so it was set adrift, seeking a home.
I believe that home is now within the much broader subject of computational services provided by ITC or computer science.
So naive student of Newyon failed to develop the tangent and Circle vector components of a point in space. They did not know how to use this more complex reference frame. Objects as particles move in such ways that there motion can be broken into component vectorsI:3 orthogonal directions(Cartesian, but generalised coordinates were also included) and a circular component with a tangent and centripetal and centrifugal components. The tangent never represents a uniform motion. The notion of uniform motion implies there is no cirtcle component and the tangent does not needto be interacting with the othe 3. No circle component means no centrifugal and centripetal components.
The circle can be completely characterised by its radius and by its tangents to points on the extended diameter line. Appolonius describes the circle in this way because it can then be generalised to the sphere and it's cones and the diameter plane. In fact the circle or sphere can be parametrised by these elements, including chords or chord planes.
While trigonometry for the inside of a circle is what we usually get taught, there is a trigonometry for the circle or sphere from the outside. Apart from Appolonius, the best modern treatment in my opinion is Norman's . It lacks one thing:?the circle component itself.
The realisation that the circle component is missing only dawned on me when I attempted to create curvilinear coordinate systems. In fact Norman claims to despise the angle, which very loosely is based on the arc. He specifies the spread between 2 lines protectively, which in that case is using trigonometric ratios of the right angled triangle. By using Appolonius projective geometry he is able to generalise this to properties called duality or perpendicularity between points and a line.
Norman's notion of duality is really a construction product. The construction product he calls the dual can be recovered by Euclidean geometric means. But it becomes clear that the observer has to impose stricter rules or definitions than Norman has to to achieve this result. In imposing these stricter rules the observer is gaining nothing, and is in fact limiting his/ her ability to cope with a broader range of relationships. It is these subjective restrictions that Grassmann realised were making geometry and Mechanics so different from one another.
The artisan who developed mechanical skill was able to utilise a much broader range of relationships in space than a classical geometers was taught to rely on! I have even read where such classicist complain of Archimedes using neusis to solve the trisection of a rotation problem ! These games were not fit for purpose, and yet they were used to impose disciplines on students that stifled as much creativity as they provoked..
An engineer or mechanic can solve this problem in an instant. They can even develop solutions to whatever accuracy is required. But a classicist would struggle to achieve it by proposition alone, bases on prior theorems Lemmae and Porisms.. The classicist would know it can be solved mechanically, but what he would not know is whether theere was an invariant method that gave the solution, and therefore could be given the title Theorem, which means handed down by the gods!
Pragmatists do not wait for the gods to reveal their secrets. They attempt to resolve the issue through human empirical skills and tools. These methods of Mechanics were highly prized but still looked down upon by philosophers who sought to establish connections with the Gods.
So, in this regard Apollonius revealed how each form had its own secrets. There were no general rules to be found but much similarity and analogy. It is the Analogy of perpendicularity which revealed a connection to the cone for a set of curves that were known as the Conics.. More importantly, the shadows cast by rigid forms revealed whether the source of light could be represented by rays of straight lies or families of parallel lines.
These linear projections carried over the curves with them, making it possible to think of a lineal relationship that could describe all curves. This was the notion of differential or Fluxions and fluents tha Newton explored. Fluxions were instantaneous descriptions of curves. Fluents were there derivatives which were also curves. These relationships were hard to convey in the terminology of his day , so Newton invented new terms.. Later, the number theorists in arithmetic objected to these terms because they did not accord with any notion then parlayed.
Newton objected to infinitesimals. They were not quantities or numbers in his view. But he could use the logos analogos method to demonstrate that rational configurations existed to quantify his geometrical insights. He never went as far as to say that the undefined notions of infinitesimals were sound. Apparently that had to wait until 1960 for some rigorously defined notion o become accepted. The question is, is it a pragmatic definition?
In any case, the notion of a circle was supersceded by the notion of infinitesimal "lengths", that is lines or tangents bounding the circle. This clearly is pragmatic but overly complicated, when a simple compas can draw a circle smoothly and mechanically. The complication becomes clear when one tries to produce a sphere in this manner. Spheres re best produced by cylinders and cones.
Let us return to the Newtonian frame of reference. It is represented by a globe on which a local Cartesian reference frame sits along the tangents and the radius. . In this configuration we have instantaneous impulses aligned with fixed local and relative reference frames. By establishing the constraints any motion could be tdescribed, and every point described in 2 ways , one based on fixity the other on dynamic instantaneity. Everything was relative.
It did not suit Newton to leave deity out of the picture, and so he rested absolutes in the deity without any hesitation or fuss.. He also pointed out that we could never determine where such a position was because of the fractal nature of reference frames, but that the best indication of absolute was within the circular reference frame because of the unique centre. However in such a frame every point becomes a centre of rotation, and it is only by measuring how much an action deviates from what one should expect from his local bucket spinning experiments, that one might be able to determine ones true position.
This is in fact as vain a hope as in the case of parallel reference frames, as the local centre may in fact be giving a result affected by some other axis of spin and so on. Even in a circular reference frame very thing remains relative, ven for god. This was Einsteins impiety!
The point here is that Newton fully expected circular vectors to be in place, and these could be resolved into instantaneous or infinitesimal vectors of the duration of an impulse. For an object to maintain a tangential course to a fixed position on a fixed globe requires some constant and unimpeded force! But in this set up the likelihood so that impulses will be encountered to deviate from this fixed uniform motion . Thus Newtons first law regales the reader with this description of the environment of space into which he is just about to enter. This is Newtons so allied inertial frame.
We can see that gyroscope, far from being alien to Newton as some thought, was in fact central to his doctrine. From his initial mention of the spinning top, to his infinitely long cylinders in fluid resistive motions, Newton was aware of gyroscope in the notion of circular and cylindrical motions analysed in instantaneous complements to these motions.
Why was he unable to make a full determination of fluid motion? I believe it was because he set up an bsolute frame and so expected to find true forces and motions, whereas he could only find relative motion equations. In particular he never observed how rotation imparts steady state conditions in rotating bodies. Thus turbulence hich was visible and expected was not always random. It often formed steady vorticular systems as Descartes surmised without real empirical evidence.
Newton could not find this with the motion and concepts he was using and retreated in some confusion. Later fluid mechanic. Helmholtz in particular with Lord Kelvin were able to set up tools and notation to describe these steady dynamic states.
The discovery of Arago was more important than Oersteds, for it shows that a current flowing in a wire is a nonsense. Instead a magnetic field rotates around a copper disc and entrains a magnetic field rotating round a steel pin. The rotation of a magnetic field was anathema to theorists of the time, who fell into the easy trap of static equilibria. Dynamic equilibria were beyond their thinking for something so simple as a bar magnet, or a piece of lodestone. And yet the liveliness of the attributes is plain to all.
It did not help when the common experimental practice was to wait for a dynamic system to settle down into a steady state before taking measurements. Everything was done to deliberately represent a system as static, when it was clearly dynamic. Rotating fields have been the real nature of magnetism from the start, and it is these rotating fields that have been confused as electrostatic phenomena.
The increasing use of vectors to establish reference frames, and their unsubtle use of affine Geometrie, confined theorists to complex mathematical models. A rotation could be described by a varying pair of orthogonal vectors. These were the centripetal centrifugal axis set in an orthogonal pair. While we can attribute this to Wallis's conic formulary, we would be mistaken to attribute it to Newton. Newton used an orthogonal pair, but they were instantaneous. The pair he used the tangent and the centripetal and centrifugal axis centred on the circle perimeter, are most apt for describing natural force systems, especially dynamic ones.
Newyon's more subtle set of vectors were not confined to lineal algebras in the affine or projective sense. They include a third type of projection and that is circular projection. With this comes circular translation, and the whole system becomes a trochoidal set of vectors.
I therefore define a vector as a " straight" stretch between 2 points or a circular stretch between 2 points relative to a common centre or a lineal combination of both if the circular stretch is recognised as a more complex action than stretch a followed by stretch b. rather it involves finding the common centre and then stretching around an arc relative to that centre.
We can combine these 2 forms of vectors if we allow a point centre at infinity. Then we can specify a parameter called curvature which determines the kind of stretch done specifically.
I am not sure I want to call them vectors, and I have used the word Twistor before. A twistor has a radius of curvature which then determines the geometrical form of the stretch. These more unwieldy formal symbols actually make better descriptors of dynamic situations. With them we can better model and understand gyroscopic behaviours in their most general form, and model as one behaviour electricity and magnetism.