One of those unexplained things you swallow as a student of Maths and Physics, a moment is simply defined as the product of a distance by a force perpendicular to the distance. However, as always this is not completely true,
A moment begins life with the greek philosophers who did not have multiplication as a concept as we do, that is a number aor arithmetic comcept. In fact many Renaissance authors would concur that they altered the original writings for their own purposes. Thus the invwntors of Arithmoi, the pythagorean school, would be mystified by arithmetic as we now believe and teach it.
The stoikeia of Euclid are introductory texts to the Platonic Philodophy of Ideas, but Plato was an ardent Pythagorean, as wellas dutiful to Socrates. This curious mix sustained a school or Academy of Philanthropist philosophy for Millenia, while contributing to the notion of a Utopian state run by the Philosophers not politicians with their knives!
The Mathematikos Eudoxus, taight the Platonists how to use a Mosaic to apprehend the experience of consciousness. In among those teaching, in Book 2 the moment is introduces together with the Gnomon. The moment is the segmented line, from which, by rotation a rectangle, that is an orthogonal parallelogram may be formed. Any other rotation than an ortho rotation produced a form called a parallelogram.
A moment therefore was specific to the ortho rotation, but any rotation of this segmented line produced a rectilinear parallelogram.
The gnomon was next of interest becausethe diameter of the parallelogram when cut by parallel lines created a form in which a similar parallelogram existed about the diameter. The other 2 forms were dual and about the parallelograms . these 2 with any specified parallelogram in the figure created a gnomon. This was a rectilinear version of the Logos Analogos analysis in the plane distinct from the more genral logos Analoos in the circle.
Thus ultimately, moments were related to the 2 intersecting chords in a circle that defined the extreme and mean ratios.
The notion of multiplication does not even enter the definition or discussion.
Later in Book 5 Eudoxus develops the logos Analogos analysis to deal with multipleforms in the Mosaic. and it is there we find the true basis of all Renaussance mathematics so called. This inevitably led to the renaissance invention of the concept of Number, and the moment became fixed with the concept of multiplication.
To Newton this was a travesty. He would not lend his name to Arithmetic, so called, and scarcely to Algebra o called, which he preferred to call Analysis.
Now let us attend to the Newyonian moment. We have of Sr William Rowan Hamilton, that the term moment may be taken as of time, by which e develops his Science of Pure Time, a development of the so called real numbers. It is rather to be thought of as the explanation of all Arithmetics within the proper spaciometric context, in which the step in space in any direction is made an analog of time progressing in one line. The inherent successional order of this arrangement conducts us to some sloppy consideration of replacing ordinal numeration with so called Cardinal numeration. Thus the die is cast for the invention of Cardinal numbers, divorcing them from all spatial considerations. We're they living things they would have died from such an operation, and indeed they are dead bones of the living thing, that being ordinal and successional quantities that self arrange!
But Hamilton gained this opinion of Newton, who was at pains to explain the need to measure in order to know of that which we philosophise. There are many things we know but cannot determine except we measure. Newton is at pains to elaborate on how this impacts our knowledge and terminology.
Thus he redefines several well known terms, common among philosophers , and postulates of the reader compliance in their use in his works. Thus he "warns" the reader ahead of time to be careful not to misunderstand his empirical and measured terminology! For with Newton most common terms are to be understood as measures!
It is for lack of herding this admonition that many found themselves in great perplexity! They would not have their minds and notions washed as it were in Newton's laundry room! In fact, for want of this many desired to insinuate that he sought to be treacherous and deceptive and to introduce error upon error!
It is a true disciple that heeds his masters words! Some being enamoured but still not compliant, also failed to understand him, and thus did him some harm in promulgating wrongly his " doctrine". But those who were compliant and obedient gained such a great reward in understanding that it is no wonder that he was spoken of in such Hyperbole.
Nevertheless, Newton pointed out that certain fundamentals, time, place, space and motion, we're not commonly understood by anyone , including himself. These were the things he endeavoured to apprehend by his philosophical method. In so doing he proposes several discourses on the subjects, intending to reveal how his considerations had much to recommend them in the quest to understand.
One key aspect was reverence for the ancient masters both Greek and Indian, not casting aside their insights as antiquated or anachronistic, but rather carrying forward their traditions to a better mastery.
The flexibility of terminology, that some utilise but without understanding, was key to Newtons subtlety and his advancement. Thus when considering the methods of exhaustion, he would not accept ad infinitum as absolute, that is to say as commensurable by humans! For him and Archimedes such things were beyond human powers. But he concurred in what was called a relative infinity, that is to say an approximate " infinity". Such a term strictly makes no sense, but what does make sense is an endless process which is truncated as is said by exhaustion! Truncation thus acknowledges an infinite process , while bringing it within the Human compass. As a profound example, consider the decimal extraction of 1/3 rd!
It is in considering these processes deeply, and with a lively mind, that Newton observes the moment , that is the segmented line magnitude occurring and recurring. What clearly changes is what the Greeks calle pelikelos, the apparent or outward size of the quantity, the scale of it. This consideration is found briefly in book 5 of the Stoikeioon, but in other sources that I do not currently know also.
Without a doubt, the fractal nature of reality, the iteration of simple forms was apparent to Newton and to his ancient masters. Thus the methods of comparison: logos analogos, we're derived from these considerations and have never yet been bested! Using such fundamental devices Newton was able to invent many novel and powerful methods and principles which he knew to be ultimately sound or safe.
His work in this vein plus commercial interests and Navigational interests lead him in a deep study of the binomial theorem, where he was able to apply the logos analogos in an iterative way, after the Indian methods of recurring fractions, as expounded by Wallis, and the work on the Sine tables and compound interest tables, to a recurring pattern, which had no regard as to kind of quantity or species as he called them. By this he derived the binomial series which is often termed as an infinite series, but this is not the case.
The binomial series is an example of relative infinity, in that for any finite n a truncated form of it may be written down. This, unlike some suggested infinite series which have no formulary, or terminology, the issue of convergence need never arise. This one simple observation escaped most of the early 20th century mathematicians, who following Euler found themselves in some difficulty with non convergent infinite series. Newton never strayed into such a minefield, because it was not consistent with the approaches of the ancients, and moreover, relied upon highly dubious theorems and. Demo stations..
So a moment is in fact a rectangular form derived by an orthogonal rotation in a straight line which has 2 segments thus conformble.
So far we have considered all in line with Newton, and indeed in line with the thoughts of Hermann Grassmann, who in correcting some ideas of his Father Justus Grassmann rediscovered this Newtonian train of thought.
The Moment is crucial to Justus Grassmanns deconstruction of Mathematics into a synthetic subject. His fundamental flaw was to accept the opinion of his teachers as to what mathematics was. Consequently he was directed to investigate every aspect in terms of Arithmetic, and was scuppered by the motion of Multiplication. Their is no notion of multiplication in mathematics. This trick was foisted upon the students of commerce nd thus accounting, by necessary concerns of trade. But in fact, neither the cardinal numbers or multiplication have any real history prior to the renaissance!
The motion of moment carries one back safely to the historical practice of forming multiple forms by factorisation. To get a multiple form one essentially divides. Aggregation and division is what our notions of space interactions are founded on. For this reason Justus could not form a real understanding of multiplication logically, a priori. He did point to a geometric interpretation, but he could not quite convince himself of it. However his son saw immediately a dynamic analogy for multiplication in the geometrical forms his father prescribed to be taught in Szeczin. He was in fact having a similar waking dream to Newton, who saw all magnitudes as being formed dynamically. In fact it is very likely that Justus conveyed this Newtonin idea as it was conveyed to him. Nevertheless, this Newtonin idea existed before him. Even Democritus and Leucipus consider all magnitudes to be dynamically formed.
The moment therefore was dynamic and Newton used it to demonstrate the concept of fluents and their Fluxions. Here Hermann takes it a stage further is in addressing the Gnomon as well as the moment.
The moment is commonly taken to stand for multiplication, especially in algebraic rhetoric. It is often confused with a measure called area, again a renaissance invention. Consequently the dynamics of the moment are ignored and obscured.
Newton restores this dynamic in the mst beautiful and clearest way. But it was only after his death that the method of Fluxions and infinite series was published, in which the method of moments is lucidly expounded even in the publishers preface. Betkeleys principal criticism is clearly explained, showing Betkeley to have been about some other purpose than falsifying mathematics thus far.! Berkely was a cleric, ad the demise of the clerical role in Education was his chief concern. For him, God's watchmen were asleep allowing all kinds of error to creep into the minds of men. In addition, he refuted the divine inspiration of Newton as some claimed.
We may fairly lay Berkeleys criticism to rest as far as it concerns Newton's Fluxions, for they are more than adequately ,and simply expounded, by the method of moments in the Method of Fluxions.
It is quite Ironic, that Lagrange, a great Fan of Newtons, should o directly influence both Grassmann nd Hamilton, who thereby become intuitive explorers of Newyon's methods and applications of the same. Not only the direct link to vectors but also to Quaternions and Ausdehnungs Groesse can be traced back to Newton via these 3.
Here Norman gives Lagrange's take on Newtons method of Momenys