The first time I came upon the definition of the quantity of motion I felt that in some way it embodied all the motion of the parts, particles or ” atoms ” of matter. I had no idea that it was a complex vector summation. I almost immediately looke back to understand the definition of the quantity f matter. This is when I came upon the mystery of density. However I finally resolved this as an allusion to the Archimedes principle of displacement of volume of watersllied to the Bouyancy of the immersed material less dense material floated in water and displaced enough water to balance against their weight. Items that sank displaced an equal volume to the item but that volume of water did not balance , thus the item was denser than water.
It became clear that density was an obscured force or pressure. Density pushed less density to one side and occupied the lowest position. It thus behaves precisely like the Newyonial motive force. But volume of density or bulk of density also determined the amount of centripetal force , quantity of matter was alike to the definition of the quantity of motive.
The quantity of motion is a quantitative bridge between the quantity of matter and the motive vis measure. Thus it becomes apparent that Newton leads the reader into the complex quantitative system he adduced from the Galilean principle. This principle itself was adduced from observations he made of the Jovian system. In addition Galileos observations of the planet Venus allowed him to set out a convincing empirical basis to the fancy of Copernicus that the sun was the centre of the universe.
In the Dialogo he sets out amusingly his ideas in a discourse, so that the pros and cons, the propositions and objections, the arguments and counter arguments are set forth. Here in words supported by a diagram he sets out the Galilean principle as a fractal structure of local space. For it to convince his peers it was necessary for each part of the fractal system to be absolute. This meant that the Jovian system was independent of the solar system, and so the system moved around the sun as if the parts were not drawn to the sun but solely to the planet Jupiter.
Similarly the planet Venus moved about the sun as if drawn to the sun but not drawn to the earth. Thus his diagram reveals on what principle this might happen, and this is the Galilean principle that centres of vis are absolute in there influence on local regions of space, by imparting the velocity of the centre to every part of the local region. In this way the local region behaves as a sole or absolute quantity that may be itself drawn to a larger centre.
Newton adduced a quantitative vis system based on. Triumvirate of vis: absolute vis, accelerative vis and motive vis. The absolute vis is that quantity of vis that is the total or entire within an isolated system. The accelerative vis is how that total vis is quantitatively and regionally distributed in the local space as observed by how it affects the velocity over time of the parts of the total throughout the region of local influence. The motive vis is how those parts of the system are individually and proportionally being drawn toward the systems centre. This was each parts centre tench and is defined precisely as a parts weight by Newton. Thus he enables motives to be compared empirically by a spring balance or by a balance of moments.
These motives are dependent on their location within the regional accelerative vis distribution, as well as the quantity of the part and the velocity change over time of the part..btoday we naturally call the distribution of vis a field, but in fact the idea of a field is not natural at all. In great part it is a concept due to the empirical observations, diagrams and philosophy of Faraday. Newton here shows a conceptual quantity distribution which acts on bodies within its influence as an absolute system, that is very much akin to a field concept, but in fact it based on an unspecified distribution law. Later he would demonstrate the inverse square law, which is a familiar field distribution that defines a modern field essentially.
It is logically clear that if motive is to be the weight of a part of an absolute system, that is the quantitative measure of that parts tendency to the centre,that a measure of tha quantity of motion is needed. This measure makes explicit the velocity of the part that is to be changed over time. But if the part has its velocity explicit the part of space within the system was also explicit. This part of the system he defined as the quantity of matter,
The quantity of matter is therefore and necessarily a part of an absolute system, and not a universal constant at all.
The concept of a quantity of matter is relative to an absolute system. It is a regional part of that system capable of changing its velocity as a whole or sole object especially in centriole tench, or propensity to move toward the systems centre. To quantify this part Newton brings it down to a geometrical measure called a volume. . But a volume clearly dies not model the differences between weight of volumes as measured by balances. In fact this problem, as Newton knew was solved by Archimedes using the displacement of water as a way to describe quantitatively the inference between volumes of parts of the absolute system. This difference as a measure was called density. It is a ratio,and it is a ratio of compared motives.
Now I always thought that Archimedes used water as the standard substance, whose motive is used by volume to quantify every part of an absolute system.. If water as used then it is a substance found everywhere on earth and thus makes it a good local standard. If it can be found everywhere in the universe it makes it a useful universal substance to be used within each absolute system as a standard for this quantitative purpose.
However it suddenly becomes apparent that Newton based his concept of a standard substabpnce not on water but on air!air too can be found everywhere on earth, but there is clearly a naive hope that it might extend out into the system of the earth and the moon!. Newton thus considered the earth moon system to be measurable in terms of density ratios of Air.
Considering this, I must assume that Newton expected the earth and moon to have such large quantities of matter that the air between them would be a negligible part of the system in terms of quantity of motion. Substantially then the centre tench of the moon to the earth and the earth to the moon reside mainly within the relative weights of these bodies. The effect of any intervening air would be negligible. This is important because the resistive force of air as wind was well known. Thus objections of the moon and earth being blown off course by respective winds are made implausible! Nevertheless Newton expected over very long times the motion between the earth and moon would be retarded by this air leading to the moon eventually falling from the sky!