Heat as a Vibratory State of Matter

There is long 17 th century tradition that heat was a vibratory state or condition of matter. It is found particularly among the Alchemical writings of philosophers and especially amongst the writings of Newton. However, when heat began to be studied, Fourier in particular adopted the analogy of a fluid.

It is said that he used Newtons observation that heat( a fluid) flows from a high temperature to a low temperature, from hot to cold. However since Newton expressed heat as a vibration he did not concur with this notion!

The higher state of vibration corresponded to the higher temperature, thus a higher vibratory state passes into a lower vibratory state.

There are several observations at superconducting temperatures where cold is observed to flow into heat! This seems anomalous except when you realise that it is a relative statement if vibration passes into lower vibration one may equally say that lower vibration passes into higher vibration
However empirical observation reveals the following. Heat is associated with lower densities of a material. Thus a warm front is associated with low pressure circulation cells, cold temperatures are associated with higher densities, thus anticyclones spread out into warmer lower density air bringing a chill.

We say high density spirals out into lower density, but this is cold flowing into heat! In addition we say heat convicts or radiates into cold , but give no thought to cold converting or radiating into heat.

If we look at convection in a metal bar, the high temperature Is associated ith a less dense volume of material, expanding and highly radiative . But consider for a moment that the cold material is attempting to radiate into this less dense region , that it is converting from its stays of higher density spreading into a region of lower density. In this case the lower temperature and lower vibratory system is nevertheless more highly effective in pushing a lower vibratory state onto a higher one.

The situation is not the simplified hot flowing into cold energy flow or even high temperature flowing to lower temperature, it is a high temperature low density high vibrational radiating and expanding state which is being penetrated by a low temperature high density low vibration radially contracting low radiation state”

Thus warm air does not rise, rather cold air pushes the less dense air up.

In terms of the znewtonian absolute system this makes perfect sense, as the quantity of motion , and the motive measure of the cold air has a greater cntipetency than the less dense warm air.



When first I came upon Newtons absolute time, I took it to mean that non relative time which rests with his God. But now I must examine his use of this word, it’s etymology at least from the Latin if not the Greek prior philosophy , to the translators use in the English version of his great Natural Philosophy of Astrological Principles!

The occasion of this research has been the reading of Galileos Dilogo , in part, the apprehension of the Galilean fractal principle contained therein and Newtons use of the same in his triumvirate of Vis, under the heading of the Centrioetency of force, that is the Centripetal Force.

I hope , by these means to reveal the backwards structural arrangement of Newtons definitions, and why the measure that is the quantity of motion is placed as a foundational or initial definition.

absolute (adj.)
late 14c., “unrestricted; complete, perfect;” also “not relative to something else” (mid-15c.), from Middle French absolut (14c., Old French asolu, Modern French absolu), from Latin absolutus, past participle of absolvere “to set free, make separate” (see absolve).

Most of the current senses also were in the Latin word. Sense evolution was “detached, disengaged,” thus “perfect, pure.” Meaning “despotic” (1610s) is from notion of “absolute in position.” Absolute monarchy is recorded from 1735 (absolute king is recorded from 1610s); scientific absolute magnitude (1902), absolute value (1907) are from early 20c. In metaphysics, the absolute “that which is absolute” is from 1809.


It may be seen that in this context and at the time Newton wrote that he meant to convey the idea of ” non relative” ” non dependant” and ” isolated” in regard to the observations and deductions Galileo had made of the Jovian system. Thus in this system the absolute vis was evident as a separate system from that of the earth moon system. In quantitative terms this had to be the whole or totality of that quantity of vis.

Through the telescope this sense of isolation is profound, and the sight of the moons of Jupiter and the realisation that they are ” independently ” rotating around Jupiter was the deepest possible shock to the system men and women of those days could experience!

It takes a mind that measures to see this system as an absolute quantity! In the sense that a triangle is a different absolute quantity o a hexagon, say. Only by seeing these as Arithmoi, pebbles or mosaic chips can the profound quantitative analogy be understood. For a picture may capture that which takes a thousand words to describe the extensive properties and propensities of, but that same picture cut into a mosaic gives a quantitative and relative encoding of each part in the Arithmos. Each Arithmos is as an absolute quantity, complete and entire to itself.

That such properties of vis, namely centripetal vis should be so evidently associated to this quantity or Arithmos is a revelation! However to use the quantitative apprehension with the vis some rule of relation must be observed. For example this absolute quantity of force might be divided uniformly though the isolated system. . However by this time, sufficient evidence and deduction had been gathered as to suggest that the rule of relationship was an inverse square law. So this quantity of vis was distributed so as to act differently at different distances from the centre. This notion was not absolute therefore as it depended on the absolute quantity of the system however it was measured by the accelerative effect this distribution of vis had throughout the system thus Newton called this measure the acclerative vis. It is a dependent subdividing of the isolated or indent quantity of vis in the system. Thus while encoding the same regional quantity, so that the total should be the same, the distribution of the summation should reflect the greater sum where the greater concentration of vis is and vice versa.

How might this be achieved quantitatively? This requires that there be another subdivision of the vis such that the greater acceleration is embodied in the greater part of the whole or if necessary the greater acceleration being in the smaller part, the force being small yet the time of application is longer. That being the requirement means that the accelerative vis must impinge a change in velocity of parts dependent on time! It is thus the measure of that velocity change over time. This is agin a definition of acceleration given by Galileo, but here the proportion of the system, ie it’s relative” bulk” or mass compared to he Abdolute, affects the quantity of vis in that part.

But then, should a part be moved centripetally as a whole then that part is a measure of motive vis in the following precise sense: if it is opposed by a spring or by the parts acting as moments to a balance! Thus a part or a summation of parts may now be given a motive as a multiple of some standard notive or weight, and balanced accordingly on a pair of scales.

Thus has Newton brought down from the starry heavens, by Alikening to a system of measures that quantify the absolute systems of planets by their motions over time and by their bulk in the heavens or yet within each absolute system.

I now turn to Abdolute time. This is not an observed property of the Jovian system, but rather an addiction by Newton, that if a system may be absolute, that is isolated and entire, that time might also be absolute. In fact it was necessary to use earth based times to discuss the changes in the Jovian system, yet it is evident that if Jupiter has moons then it has it’s own system for recording Tymes! That being the point explicitly made within the first chapter of genesis as regards the function of a moon!

Given then that each absolute system has its own absolute time, that makes time a relative concept! But that was unacceptable or that way lies confusion! Thus absolute that is independent time was the solution. Such a time may well exist with god, but it’s conception was more mathematically adduced than for any other reason. If such a time is not demanded how will 1 second be measured any where in the universe never mind on the planet earth.

Absolute time represents a simple knee jerk reaction to the difficulty of quantitatively measuring acceleration and velocity in these absolute systems. Later Einstein would point the way to how to relate all these relative measures together in a system based exactly on Galileos principle of uniform motion of reference frames.

Returning now to the motive, it is clearly a quantity that is proportional to the bulk or mass of a part and the velocity of a part under accelerative vis. It is exactly a quantity of motion. What we find is that Newton defines such a measure almost as the first of his definitions before defining vis or acceleration or velocity. These are assumed to be known from Galileo’s work.

The quantity of motion is thus his final definition of motive, placed first to emphasise it as a fundamental unit within a centripetally acting absolute vis system, with an accelerative bis rule.

This is. Newtons exposition of the Galilean principle in a form amenable to quantitative measures.

Newton’s Centripetal Force

Newtons definition of centripetal force precedes his application of the Galilean principle which he reduces to 3 sorts or kinds of vis as a centripetal force.the absolute, the accelerative and the motive.

It is to be noted that in defining centripetal force the first presentations describe it as drawing a body away from its natural or innate rectilinear motion, that being the propensity of its innate force. However this is rather a stronger notion than the original Greek idea of a ” good” or “straight ” line allows. Such a good line if it be true to its nature in the motion may in fact be a curve! In any case the assumption that motion is naturally rectilinear is hardly empirical by any degree!

Supposing now that the innate motion merely attempts to continuously recede from the centre which impels it from its recession hardly spoils the plot, but in fact enhances it in the many and congenial examples he gives of a centripetal force in action, or behaving as he wishes it to be described.

We may also have missed the import of his second example, that such a force is as an example magnetic, and as a third example but of an unknown force, gravitational!. So in proposing this definition it is not to define gravity but all kinds of force that tend the motion of a body continually to ward a centre, and so doing by altering the bodies innate velocity Leo, as well as its general recessive course, we’re it to be viewed from the vantage of the pulling force centre!

There is then a ballistic discussion mediated on the introduction of the idea that the velocity of a body is altered by a centripetal force, in which the relative view is again switched as the explosive power of gunpowder now impels a body to some velocity which gives it increasing range , until at last a velocity should be reached by which the projectile completes one orbit! The argument being so clear and logical that it has been remarked that it alone was sufficient to convince his peers that he knew whereof hew rote. But it should be noted that with increasing peed the missile was said to pass out of orbit to continue into gods universe unhindered. . This vision was clearly advanced so that he might bring the moon into his discussion forthwith and deny ny argument that the moon was drawn out of its way by the same centripetal gravity that governed the ballistics of the projectile.

This having been advanced he leaves it to the mathematicians( that is the qualified astrologers) to determine this exact velocity , and the intensity of this centripetal force so that any body in orbit might just remain so , falling so as to miss the earth continually, but not to come crashing down onto the earth, nor yet to fly away.

Now some have read this and failed to see the complexity of what Newton set out to do. As a consequence they have not grasped the complex nature of the orbital problem in particular, the many forces as impulses or resistive forces or impelling or expelling forces that are herein described and employed to convey the conceptual framework he thn goes on to address. Therefore ome have rejected his later more careful description of circular and orbital motion in which centrifugal force is admitted to the description. It is empirically obvious to all that such a force is needed to account fully for observed behaviours, but some have sought to make velocity the source of this centrifugal force. The changes in in the tension on a string would properly imply both a centripetal and centrifugal force, which when removed by swinging a projectile in an elliptical arc results in the projectile crashing to the ground under no centripetal orce by the string! Thus the string not only must supply centripetal force but also centrifugal force to keep the projectile in that correct attitude to orbit.. The velocity itself if it was solely rectilinear would behave tangentially in breaking loose of the projectile, but instead it behaves as if it has another velocity imparted by a centrifugal force compounded with the supposed rectilineal one ( by those who would argue that rectilinear is a straight tangential velocity instantaneous to the orbital curve).

That being said we must now proceed past the moon to the Jovian system observed by Galileo to follow newtons establishment of his force triumvirate.

Centripetal force is of 3 kinds, and each kind is a measure/ Metron or measured quantity the absolute vis measure is a quantitative appreciation of a central agency. His example is the many kinds of lodestones nd their corresponding intensity. Thus should we specify a particular lodestone, or even Jupiter, that is a particular absolute vid.. Thus absolute means an isolated or specific potency under investigation, and the sole cause of all that is considered in following discussions.. It is defined as acting from the centre.

Later Newton adds that this absolute vis is the cause or central causative agentin the system. I have taken that to mean in the universal system god, but in a local system, clearly it is a specified quantitative object.it has to be quantitative because he is establishing a quantitative fractal system as designed by Galileo.

The accelerative vis is next defined. There are more examples of how the accelerative vis varies around the specified object or centre.. Thus if we isolate a quantity from an extensive magnitude, that is only the beginning. We must now examine that quantity carefully snd specify som sub quantity measure!in other words, taking the object as a whole we must now divide it into multiple parts..in this case taking the intensity of the absolute vis we must now render it into its behavioural parts. In this case , as he later shows, he desires to establish an inverse square proportionality as the parts. This is his taccrlerative vis proportion, which he will later empirically and geometrically and astrologically establish.

He quotes one observation of Galileo directly, that accelerative vis imparts the same velocity regardless of mass at the sae height or distance from the accelerative bis centre.. But the absolute force for each identified system will be different, so the subsystem accelerative force will have a different value but the same law of calculating the measures..

Now I took the accelerative vis to be an instance of a universal vis, an absolute vis, but clearly it is a scale of behavioural action centred on the centre of the absolute vis.. Thus in the universal view accelerative vis describes how the various parts of the universe should behave relative to the centre. Immediately it becomes apparent that this system cannot have a universal application! No one knows where the centre of the universe might be!thus despite its general applicability it is not a universal system but a local empirical one.

In that regard, yhe recent expositions on dark matter seem premature. If we choose a galaxy as an object we can use this system to estimate whether an inverse square law pplirs to the galactic syste, the galaxy is assumed as the absolute vis nd an accelerative vis is designed for it from the galactic centre. When this is done, it apparently does not work! So dark matter was invented o explain this” anomaly”. This ord Anomally means infringement of alas, but it is us who imposed this law on the galaxy in the first place, clearly our law is not a correct description for galactic systems! We do not need to invent dark matter, we just need to get the correct law, which may beone of the other laws Newton tried throughout his astrological principles.

Finally he describes the motive vis measure or Metron. This basically allows the observer to sum up the behaviour of masses or parts of a system which are being impelled by the accelerative vis to move toward or about the centre. In very particular, at the same level one mass may be blanced on a weighing device against not her. Thus the motive vis measures parts against parts under the influence of the same accelerative vis and uses the balance or ahookes spring to establish proportionalmeasures like mass, weight etc.

Again Newton relates this to the astrological Jovian system, in which the moons might now be weighed against each other within the one accelerative force of the absolute Jovian system

There is no as yet direct statement that the accelerative force if the centre itself is in motion imparts this additional motion to the parts it. Accelerates, but this will be advanced later.

The Galilean Principle

The Galilean principle is mentioned in passing. That an object within a carriage sealed from visual contact with the outside will be seen by an observer in that carriage to move as it should. Thus it is indistinguishable Whether the object is in a moving carriage or not.

The movement of the carriage is conveyed to the object and to the air within the carriage! We must take this further. Whatever is within that carriage is moved in concert with the carriage, thus whatever medium or aether or property by which some physical power exhibits itself is also moved.mthus we say the laws of physics hold in any uniformly moving frame of reference. By this we mean that that motion of the frame is imparted to any phenomenon within that frame.

How then do some say and teach that the speed of light is a universal constant? That nothing travels faster than the speed of light? It is clear, by the Galilean principle that light or any object travels at its speed plus the speed of the frame in which it is emitted!

The question is can we observe anything moving faster than light? Clearly not, but what we can observe are the Doppler shift effects due to portions of a light beam ( or a sound wave) crowding in to tHe medium of transmission!

What if the medium of transmission moves independently of the carriage? Then we would expect some Doppler like variations in the light signal. At sea level Michelson and Moreley found no variation. However at higher altitudes scientists have found perceptible variations! At those heights and densities the relative motion of all mediums is not sufficiently coupled to the earth to fulfill Galileos principle.

The speed of light is not an immutable constant, but it is limited as a bulk property of any medium. Thus Lorentz contractions or Doppler shift effects do have to be accounted for in measurements using light or radiation, not because nothing can travel faster than the speed of light but because we cannot observe anything travelling faster than light speed.

Suppose now we make a measurement and by calculation it comes out faster than light speed, the question is how have we “observed” that? If we have not accounted for light speed limits in a material correctly we will make an error. For example if I measure light through different media I get different speeds . How is that constant? The assumption of a constant or maximum speed comes from a bulk property analysis. It is assumed to be maximum in a vacuum. However what if it is a maximum not in a vacuum but in some other material? How will we know?

Finally what if some other phenomena is actually faster than light, but can be detected by instruments that do not rely on light or electromagnetic radiation? Neutrinos are posited as neutral electromagnetically. If they travel faster than light and we use a statistical average detection method rather than direct observation we may very well ” detect” faster than light statistical averages!

Some investigation is in order rather than dogmatism.

Trisecting an Acute Arc

Draw a circle radius r and extend its radius to 3r, by marking off r 3 times. This creates a ruler
Bisect the ruler to find the centre for a semi circle radius 1.5 r.
Draw an acute arc on the small circle and extend the side containing the arc to cut the semi circle.
Create a right triangle by these means in the semi circle. Draw parallel lines to trisect the sine length of the right triangle and extend these to the semi circular perimeter. Connect these points on the semi circle to the centre of the small circle. Where they cross the petimeter of the small circle is a good approximation to a third of the given arc.

This uses the angle subtended by an arc at the centre of the semicircle is twice that subtended at the petimeter

This approximation is less accurate the nearer the arc gets to a quarter arc, and better the nearer the arc reduces to nil turn. It is good below a 1/6th turn.

The method can be improved by using more sections, and indeed the method allows us to section an accute arc as many times as we like with increasing accuracy.

The behaviour as the arc approaches a quarter turn and as the sectioning becomes very large is interesting. There is no angle that is small enough to exist between a tangent and its intersection point on a circle! This is a proposition in the Stoikeia book 3. Thus we can not find a section that at the same time as it approaches the orthogonal radius extends outside of the circle without cutting it in 2 places.

The circle thus bounds an infinitely divisible process . The accuracy cannot be expressed in a finite ratio. When a finite ratio is found it is a special quanta and we define it as uniform, but it actually is beyond our ability to empirically determine.

Pi is transcendental precisely because it is beyond our ability to resolve it.

Pi, I and the ratio of the perimeter of a semi circle to its diameter represent the concept of a transcendental ratio, a ratio we can state but not commensurate.

Drawing a arbitrary angle and then trisecting it’s ray segments allows a triangular arallel grid to be drawn. The grid has interesting roper ties. First it houses an arc that has a ratio to a larger arc 1:3 . But the larger arc is 3 times the rotational displacement of the smaller. Thus 3 of the smaller arcs can be topologically transformed to the larger arc.
Finally three of these small arcs nestle in a close region near the large arc within an array of rhomboid forms. These forms are transacted by the large arc. This indicates that by moving the next part of the trisection to this area may give a highly accurat pair of intersection points for trisecting the larger arc. These drawn back to the angle vertex may give a good approximation to trisecting an angle.

The fascination of attempting to proportion an arc by the radial proportions and parallel lines and circle theorems , albeit approximately has led me to attempt a kissing circles design! Using the arc subtending an angle at the permeter theorem I attempt to section an arc into 6 pieces, roughly equal, by by using a kissing circle of r and 3r , and positioning the angle to be trisected at the kissing point. Sectioning g the diameter of the 3r circle into six I use parallel lines to the angle at each segment to section the arc of the 3 r circle. Using the last 2 I subtended an angle at the kissing point of approximately a third of the given angle.

Newtonian Fluid Motive Revisited

From time to time my system does an evaluative reboot.

Those who have read my expressions from the beginning, many years ago will recall that my modes operant is to dump thoughts to prevent them clogging up my system. Thus all my writings do not amount to any particular settled view but merely the last opinion or expression of opinion I was able to dump as elegantly as possible!

Unfortunately my writings are full of typos because I rarely retread to correct, unless someone has expressed an opinion on a piece of my garbage.

Long ago I recognised the ephemerality of opinion or drawn conclusions. Both rely on a process of reasoning in which if the basis is in error then so is the conclusion, and knowledge is always in error or rather, approximate in nature. So it is with my expression of. Newtonian Motive.

I have long since grasped at a causal relationship between ,otive and acceleration, based upon one sentence in which Newton ascribes a congruence relationship to motive and acceleration as to celerity and velocity. . I am no Latin scholar and I read parts of Newtons description from Latin versions of the Principia , the Astrological Principles. However many competent English translations now exist on the Web , and it is in perusing these that I find my understanding to be defective as to Newtons expression and intention, allowing for the translators intention!

It is clear that more scholarship has become freely available in regard to Newtons manuscripts and a greater freedom to critically analyse his works, notes, investigations, alchemical, musical and philosophical. No longer are we subject to a heavily biased account of his life and views. Scholars today provide a valuable and often valid insight into his thoughts, life and Times.

I have laboured to shake off the fairy toes and hero worship that were fed to me as a child, with little real justification. Yes indeed a child’s mind cannot grasp many adult concepts but to deliberately insist in a lie, rather than denoting a myth for what it is is tantamount to using statecraft and propaganda on impressionable young minds.

So it is that I come to look again at the words of Newton as translated to meditate further on his rich conceptions, and to uncover what it might be that Shunya has revealed of itself to him.


The flux moment of a rectangle.

It was Berkely that drew attention to the general problem of misunderstanding Newtonian concepts particularly that of Fluxions. At the time of Newtons writing of the Astrological Principles Mathematics had barely become a subject in England and the Ameicas thereafter, and so many found Newtons advanced geometrical demonstrations very challenging.

The word moment Newton refers casually here as mutation, but in fact its older use was in terms of motion of an instantaneous sort.thus it was associated by Archimedes with the balancing of weights on a rod, whereby the moments of turn or rotation oppose and cancel each other. In this important use of the term Archimedes demonstrated how the rectangle was an important measure of balance.

However the rectangle was constantly changing due to rotation of the arms of the balance, and thus the moment mutated the rectangle unless counter balanced.

How does this mutation change? The idea in Newtons word were that the rectangle was subject to a flux in its defining sides. This was an unfamiliar concept and observation. Berkely had certainly never seen or noticed it. However Newton had and attempted to explain it mathematically. For the reader needs only to consider the rectangles involved in a balance or pair of scales.

To be sure many readers will not recognise the rectangle as the general product or multiplication of 2 quantitiesA and B, and this is a fault in geometrical education, which also lead to misunderstanding. So the 2 quantities in question for a balance would be the weights and the distance from the point of balance. The rectangle formed is by constructing a rectangle whose sides coo respond to these 2 quantities by a length correspondence..

The rectangle mutates as the unbalanced arm swings . For these purposes the changing pivot distance is measured not along the arm but perpendicularly to a vertical line( in this card) through the pivot point. In this case the moment changes in only one of its quantities.

Now considering a mutation in which both quantities change is still considering the moment of a rectangle under these condition. Newton selects the word moment for this derivation because this is an important result in the most general form. Any moment can be calculated from it.

So what is flux or mutation? Newton sets out a simple case where the length A is in flux. His case is simple in that A does not change its value. Suppose A to be the length of a stick, then as that stick moves in flux or flow it length is assumed not to change. What changes is the position of its centroid.. However Newton does not first derive its centroid. Instead he subtracts a portion from the ” bottom “of the stick and adds it to the “top” of the stick. This is a curious way of thinking of flux, but in fact it is used everyday in nimation and particularly computer animation..

The stick thus moves along its length without compression or contraction. This seems to be the ordinary flow of the motion of an incompressible substance of a finite quantity..
Suppose now this were to occur in the 2 defining directions of a rectangle! Intuitively we would expect the rectangle to split or to move as a whole in some direction between the 2 sides. The second option would describe a rectangle in flux or flowing in a fluid., the first describes a rectangle transforming either by splitting or ” morphing” into some other rectangular form

Berkely clearly does not consider this, but rather clings to the word augment. Thus he ignores the general condition or constraint of flux or flow, replacing it by the concept of augment. In his concept augment allows a stick to grow for example, and there are no,fixed quantities only initial values.

If anyone has ever had the dubious pleasure of ” completing ” the square in solving a quadratic equation you will be aware of the somewhat seemingly odd behaviour of the signs. This behaviour becomes even more confusing if you consider a negative case! However Newton understands this behaviour more thoroughly than Berkely who does nt even consider the case! It is known that at the time many mathematicians had a loathing for the negative integers nd rational numbers nd a deep shock and foreboding of the ” complex” numbers. Newton had no such issues.

So Newton calculates the Change in the product if it is in flow. In this case as soon as the bottom decremnts the top increments thus to find how this change effects the product the simple comparison is between the decremented product and the augmented product.

The result is inevitably as. Newton stes it.

Reading a page further back reveals Newtons conceptual and notational framework explained briefly.


Firstly it is fundamentally important that a quantity is derived or formulated as a product of diverse parts. This he calls a genitum. He lists this product nature: multiplication, division,extraction of roots, proportions of various sorts etc..

He defines quantity as being of finite magnitude, or rather any quantity is a finite magnitude. A magnitude is an experience of extensivenesses and so a quantity is a finite bounded experience of the magnitude. Now these particular quantities are conceived to be in flux, and thus these magnitudes are dynamic in nature and essence. The quantity thus is conceived as flowing from some nascent finiteness!

Here is where Fluxions differs from differential geometry as taught and developed by the Leibniz school. Newton based his concepts on a nascent or evanescent physicality! A genitum as a product had to generate its quantity. Newton gave several general types likened to velocity, flux and his own notions of Fluxions, a particular instance of the general or common meaning of a Fluxions.

It is to be sure that Newton never doubted that Fluxions were perceptible at some indeterminate stage he called nascent. Beyond that no product value was assignable. Thus Newton recognised what is known as the principle of exhaustion as ” limiting” the applicability of Fluxions. Berkely in attacking this insubstantial nature shoots himself in the foot. He being a cleric every Sunday proclaimed the reality of these insubstantial things ! However he chose to attack them as numbers, which forever estranged number from magnitude! There was no healing of this division until The Grassmanns developed the Ausdehnungslehre.

Genita as products are the initial nascent expression of a quantity as a magnitude. They are where our analysis stops and synthesis begins. They correspond precisely to the Euclidean seemeia or indicators.

The product form is vital. Of the many processes in geometry, the product process is the most dynamic. In that process forms are constructed in multiple formats, and factors and transformations are identified nd utilised. The process of induction to develop or calculate a value is well used and underpins a fundamental iterative process first expressed in Euclids factorisation algorithm. Recursion or iteration are fundamental to developing certain Arithmoi, and when Newton refers to arithmetic it is to a branch of geometry that he draws attention to.

So how can he employ Arithmoi to the description of dynamic flowing behaviours?

The Concept of Differentials was pioneered by DesCartes and was a standard analytical tool in Newtons day. Leaving aside DesCartes ” plagiarism” we nevertheless acknowledge his genius role in the reconstruction of Geometry. However it is Wallis no popularised the concept of geometric algebra, or Algebra as he called it based on his reading of various Islamic texts.

We must also mention Bombelli even above Cardano nd Tartaglia, Vietnam ,Hartiot and others. There is a rich historical background that Newton was able to research through as directed by Barrow and others.

So the differential was a key instrument in sparking Isaacs development of genita! What he excludes from genita are precisely those forms hich unduly differentials. Genita being undifferentiated in this precise way served as models of flux or dynamic quantities. Where do they come from? Where does nothing come from! In Newtons day it was tacitly assumed that they originated in God. Today it is acceptable to state that they are Fractl in origin and nature and exist at ll scales.

While the Pythagorean goal was a sunthemata, a covenant with a deity for which sum bola or visible symbols existed as a reminder of the contract terms for both parties, today most scientist work for Mammon and the Praise of society. A fractal basis restores the deep abiding mystery and magical beauty of dynamic space .

Newton defines a moment as the momentary augment or decrement of a quantity, but warns something that Berkely ignored. A moment is a dynamic flux. It is not to be identified with any particle it gives rise to either in augment or in decrement!it is also not the first augmen/ decrementt but the difference of any 2 augment / decrement combination.

The concept is tricky enough to perceive as a dynamism. Newton thus defines his conception of a flux. His concept requires that both augment and decrement be accounted for simultaneously!

One can query why he made such a choice, and why other formats were not better, but the fact is he chose to define his concept in a way that covered the positive and negative arithmetics at once!

In presentation it must be acknowledged that Newton was capable of great sophistication, and often mystified his peers, but it should be evident that in this case Newton found a way to deal with the general issue ith out having to multiply case upon case due to arithmetical differences!

While it has been explained how Berkely did not understand the concot of physical flux, it must also be said neither did many of Newtons own students.

In Book 2 of the Stoikeia one will find the general geometric principle on which Newton based his regime. In this proposition a line segment is increased by a given segment and a proportion is stated in relation to this. It is this difference that Newton calls the mutation or moment of a rectangle.

The remarkable point to note is that from what are essentially constant forms a dynamic measure is derived! Newton of course adds a bit of background conception, but the geometrical position is unassailable in local Geometries. Relativistic or hyperbolic Geometries may fault the measure but that is another issue.

Thus without the paraphernalia of limits. Newton establishes a geometrically sound dynmic measure based on genita, the principle of exhaustion and iteration or recursion of an ad infinitum nature. In such a context it is reasonable and required to accept approximations as being good enough for pragmatic purposes of measurement. Also , like a lense various views of a flux are possible at different scales, and this is the view that Taylor and Maclaurin explored in their expansion theories.