Newton, in his opening remarks cites Pappus as the source of Mechanics in his time. It is known also, the Mechanics and thus also Geometry was considered an unfit subject for a classically trained Scholar to major in. It was fit only for those who would adopt the pragmatic roles of Engineers and Architects and artisans.
In addition, Newton adduces the tendency to move away from figurative geometry to a Mathematics based on Laws. These laws being the so called Laws of Mathematics.
We see here also a pedagogical snobbery. That such Laws could be said to exist requires that Mathematics exists and has immutable laws. Tracing back Mathematics to determine such laws does not help to illuminate said laws. Instead one finds that the basis of Mathematics only goes back to the 1500's where the term was adopted by Medieval monks to describe that which those who were Pythagorean Scholars studied as Astrology.
Beyond that we meet up with the Quadriium which every able student was directed in after studying Aristotelian, grammar, logic and Rhetoric. Alternatively, through the Neo Pythagorean school the works of Euclid and Archimedes and Apollonius were studied . We may go back further to the Pythagorean school in Italy who taught publicly, but also privately to those deignated as Mathematikos, dedicated students devoted to the Astrological studies of their masters.
By this analysis we may securely determine that any laws of mathematics are in fact laws of Astrology, and the " moderns" as Newton calls them, have in fact described these laws in Algebraic terminology, as far as they can.
Newton stated that books 1 and 2 were mathematical while book 3 of the Principia was philosophical. He used theorems derived in book 1 to support his philosophy of the system of planetary motions . This is precisely why mathematikos are Astrologers! We appear to have lost that Pythagorean link in our understanding of Mathematics. Newton distinguished physics (Phusis) from mathematics because Phusis requires a dual of opposing actions , at least in the ancient Greek texts on the subject. Mathematics requires careful methods of thought and comparisons of records to determine a position or relation between bodies. These relations, if general may be thought of as laws, but it is a big step to go from these laws of Mathematics to considering them as God's laws for ordering the behaviour of planets.
In fact, the mathematical laws only went from models of the behaviours to laws of the motions when they accurately and repeatedly extrapolated to accurate predictions of future eclipses, etc.
Since the ancients (as we are told by Pappus), made great account of the science of mechanics in the investigation of natural things; and the moderns, laying aside substantial forms and occult qualities, have endeavoured to subject the phænomena of nature to the laws of mathematics, I have in this treatise cultivated mathematics so far as it regards philosophy. The ancients considered mechanics in a twofold respect; as rational, which proceeds accurately by demonstration: and practical. To practical mechanics all the manual arts belong, from which mechanics took its name. But as artificers do not work with perfect accuracy, it comes to pass that mechanics is so distinguished from geometry, that what is perfectly accurate is called geometrical, what is less so, is called mechanical. But the errors are not in the art, but in the artificers. He that works with less accuracy is an imperfect mechanic; and if any could work with perfect accuracy, he would be the most perfect mechanic of all; for the description if right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn; for it requires that the learner should first be taught to describe these accurately, before he enters upon geometry; then it shows how by these operations problems may be solved. To describe right lines and circles are problems, but not geometrical problems. The solution of these problems is required from mechanics; and by geometry the use of them, when so solved, is shown; and it is the glory of geometry that from those few principles, brought from without, it is able to produce so many things. Therefore geometry is founded in mechanical practice, and is nothing but that part of universal mechanics which accurately proposes and demonstrates the art of measuring. But since the manual arts are chiefly conversant in the moving of bodies, it comes to pass that geometry is commonly referred to their magnitudes, and mechanics to their motion. In this sense rational mechanics will be the science of motions resulting from any forces whatsoever, and of the forces required to produce any motions, accurately proposed and demonstrated. This part of mechanics was cultivated by the ancients in the five powers which relate to manual arts, who considered gravity (it not being a manual power), no otherwise than as it moved weights by those powers. Our design not respecting arts, but philosophy, and our subject not manual but natural powers, we consider chiefly those things which relate to gravity, levity, elastic force, the resistance of fluids, and the like forces, whether attractive or impulsive; and therefore we offer this work as the mathematical principles if philosophy; for all the difficulty of philosophy seems to consist in this—from the phænomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phænomena; and to this end the general propositions in the first and second book are directed. In the third book we give an example of this in the explication of the System of the World; for by the propositions mathematically demonstrated in the former books, we in the third derive from the celestial phenomena the forces of gravity with which bodies tend to the sun and the several planets. Then from these forces, by other propositions which are also mathematical, we deduce the motions of the planets, the comets, the moon, and the sea. I wish we could derive the rest of the phænomena of nature by the same kind of reasoning from mechanical principles; for I am induced by many reasons to suspect that they may all depend upon certain forces by which the particles of bodies, by some causes hitherto unknown, are either mutually impelled towards each other, and cohere in regular figures, or are repelled and recede from each other; which forces being unknown, philosophers have hitherto attempted the search of nature in vain; but I hope the principles here laid down will afford some light either to this or some truer method of philosophy.
Theb5 manual powers are what ?
The 5 powers appear to be wind, water, man or beast , falling Earth ( gravity) and melting heat ( Fire)? These were all made use of by the Artisans of ancient times to cause mechanical motion.
It is important to note that Newton characterises fluids as resistive mediums. While this is a small point at this stage, it shows itself up in his confusion over fluid forces. He does not assign motive in a positive sense to fluids. Thus fluids are inherently reactive and resistant, and therefore not transmitters of his concept of motive. By simply changing this opinion one can see that motive becomes an analogy for pressure and energy, as well as inertial behaviours. We can then tackle fluids as positive contributors of motive, but also involved with transformation of the corpus transmitting motive.
The only way to apprehend this is by instantaneous description of spatial distribution , with that instantaneous " slide" changing over time. The motive then act not only in space but also through time .
The notion of spacetime officially structures this set of relationships. Fluid dynamics demands a spacetime algebra.
There are 2 other philosophers who share Newyons view: Hermann Grassmann and Sir Illiam Rowan Hamilton. Th ubtlety of. Newton is reflected well in Hamilton's writing style, while the philosophical concerns about Mechanics as the source of Geometry are well represented by Grassmann.
It is these 2 who actually develop Newton's view of mathematic and physics, not as subjects but as processes of natural philosophy. The academic studies of Hamilton brought hom face to face with Newton's style of education and direction of tidy, while the synthesis concerns of Hermann's father Justus and his strict direction, served to lead Hermann into fundamental insights in the mechanical aspects of ancient Greek constructions as manifestations of an inherent 3d geometry of motions, in other words Astrology, by another name.
By their time, in distinction to Newton's, geometry had become a distinct subject boundary, and was already being mistreated as inconsequential to Algrbra. By this time,Hamilton had become a champion of Algrbra as a subject, while Grassmann was developing the more fundamental group and ring theoretic Algebra of combinations. Contrast this with Newton who said Algebra made him nauseous!
While Hamilton pursued the academic line of progress, Grassmann pursued the research line of regression. He went back to the fundamentals, and resynthesized the whole of Mathematics, he thought, after his Father's aims, on a strict, rigorous group/ ring theoretic basis. Because it was so strict, Grassmann's notation meant something entirely different to standard notation. Hamilton, in his theory of couples devised the same kind of notation, but used it academically, not fundamentally.
Hamilton's aim was to justify the status quo including the imaginary magnitudes. Still in his day people understood the difference twist magnitude and quantity. Hermann's aim was to reconstruct, resynthesise and re present on sure and secure foundations a newly revamped mathematics. Consequently his conception went back to Pappus and to Spollonius and even to Euclid, as presented by Legendre. Hamilton academically attempted to blaze new trails by basing his conception on Pure time.
Thus Hamilton and Grassmann both dug deep into the structure of ancient Pythagorean and platonic philosophical descriptions of forms in reality, but one more solidly the other more aethereally.combined they more closely matched Newtonian thinking than any others.
We can add to the strong base laid by the Platonic Pythagoreans as opposed to the Aristotelian Pythagoreans, to coin a distinction among the Platonic Pythagoreans, the recent work of Norman Wildberger et al. Such work makes up for a deficiency ven in Newton's conceptions, but are themselves refined by Newton's conceptions once order is established. By this I refer to the primal importance of Mosaics in establishing the mind gifted by the Musai.
Such a mind is fit for all forms of aesthetics and Astrology, and was highly prized by Plato, and therefore Aristotle.. It was the mind of all Pythagorean Scholars who had achieved the status Mathematikos. It is clear that Plato did not achieve this grade, neither did Aristotle. It is possibly Euclids Pythagorean Platonic school that could be granted this Accolade, but I have yet to find evidence to bear this out.
Eudoxus, who joined the communal life at the Academy possibly was at this level, but certainly he does not seem to have played upon it, a kind of equality and fraternity bing valued in the academy among the seniors.
The Academy and the Lyceum have distinct histories and contrasting influences. It is the Lyceum that shapes the Arabic educational system , but it is the Academy that shapes the Neo Pythagoreans up to the fall of Roman Greece. Thus Pappus and later proculus draw upon the tradition of the Academy, but later, medieval scholars draw upon the Lyceum tradition through the islamic scholarship.
It is these 2 traditions that Newton had cause to consider, and from which he amalgamated his Astrological ( Mathematikos) principles for Philosophy. It is from these 2 traditions that he derives his Mechanical views on Geometry, not as a separate subject, nor indeed to be separated out as a separate subject, but rather as the shining facet of a many faceted jewel of Mechanics.
This crystal passes through the Prussian educational system to the Grassmanns, while the facet passes through the British education system to Hamilton, who at once reflects its brilliance into the prisms of the couples and the triples and then the Quaternions! The significance of its Pythagorean origin I'd not escape his keen scholarship.
Meanwhile the crystallographic implications of this jewel did not escape the Grassmanns!
Follow this playlist to see the fundamental mosaic concept.