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.