We appear to have moved on, but we have not really.
design obscures the basic principles or it enshrines them. In these examples i illustrate how the fundamental notion that of periodic intensity change , comes through in various guises.
The course that leads to us measuring with tempo is a long one going back into ancient histories.
We all experience movement, and we all know what movement is from a vast collection of interactions involving it. but to agree upon a common measure, that is a metron, proved to be hard, due to intransigence and cultural inertias. Movement, along with relationship are core fundamental notions of any animates psyche. Thus here is no one metron that fits or suits all for dynamic changes in intensity that are experienced subjectively by each individual uniquely.
For example, the first marathon runner who gave his life to pass on a mesage, sanctified a distance which has remained with us to this day. That agreed distance could form the basis for a metron of motion, However it is ipractical to apply it to small local motions, movement of an arm or a leg say. It literally makes more sense to use the movements of body parts to measure movement or motion. But there are several body parts that move, and so there are many metrons that can be and are used.
These metrons are exemplified in the activity we call rhythmical movement: walking, dancing, rotating, speaking, singing etc. So there is a collection of metrons for motion. This means that motion can be understood as some quantity that can be compared.
We are blinded by the quantities space. If we were blind we would "see" the other quantities we apprehend differently, movement being one of these. Collecting all these attributes of the quantity of movement requires one to accept ones experience as magnitude, and the importance of magnitude as intensity changes to denote "boundary", which though it appears to be a visual concept is in fact a strongly proprioceptive and auditory one. The isual input adds icing to the cake of boundary perception, because we know when we have crossed a boundary or come up against one by many other sensory experiences.
Thus these sequential boundary changes eventually allowed us to choose periosic change as a metron for motion, AS per Euclid we do not have to specify periodic change beyond its relationships to other attribues we wish to measure by a metron.
Tempo and Rhthm are both called time, but they derive that name from tyme used to mean the tracking of the position of a moving referent, such as the sun, moon or stars. Because of the development of water horological tools duration became an analogy for tyme, the motion track of a planet say.
From there by a process of iteraive development, the combined ideas were the guiding principles of mechanical developments of gears, armilleries, antikythera and clocks. For this purpose, time was reinvented by a complex, convoluted and tautological proposal based on ratios.All these measures of motion desperately needed simplifying, and during the renaissance this task was commenced and progressed by empirical decrees to the system of weighs and measures we use today. Time as we know it is a complex cultural invention, but it plays this role through its key involement with the metron of motion. Huygens provided the modern era with this complex version of time that links time and space through our subjective experience of periodic motion.
The quantity of motion is the measure of the same, arising from the velocity and the quantity of matter conjunctly
Like Euclid Newton lays out his systematic definitions. Again he introduces anundefined but apprehensible term, and the combinatorial relationship. The analogy is complete, but it is not between the 2 definitions, but between Newton and Euclid.
I have spent some time exploring our subjective notions of movement, This notion of the measure being defined in this way makes sense litterally because the quantity of movement must reflect By how much and how many things move. Newton elaborates on this not in definition2 but in definition one. However his explanation of it in definition2 is that the relationship is strictly proportional. it says nothing about the distribution of the quantity of matter or the individual velocities of any parts. He presents oly one example in which the object is first doubled in bulk, and then doubled in velocity. Without elaboration he moves on. The reason is clear: the quantity of motion is a troublesome measure requiring strict guidance to its application. It also require an algebra hat combines part motions into a whole, and that is a vector algebra. Thus he moves on because this definition is only the most straight forward of idea. It requires a fractal understanding to apply this simple seeming relation!
The motion of the whole is the sum of the motion of all the parts
. `in this way Newton intimates that he will deal with complex other cases by another more elaborate application of this principle and definition. It turns out to be a vector fractal treatment based entirely on Euclidean principles.
`in Newton's mind therefore it is clear that the quantity of motion may be apprehensible but not result in a complex whole moving its centre of symmetry, but should those motions align and compound each other in a common resultant direction then the quantity of motionshould result in a common velocity.
Now this is very interesting, because for a complex whole the internal motions may produce an external overriding resultant wjich describes the motion of the whole. Furthermore, should these internal quantities of motion move relative to onw another then it is possible to describe some complex resultant motion that is dependent on them.
Thus in a complex whole we have an internal economy which inheres the motion of the whole, and to which quantities of motion may be added and subtracted by collision. The resultant quantity of motion therefore may not just be a "straight " lined motion. It may in fact be a Brownian motion!
Newton in proposing this definition goes on to elaborate as may applications of it to motion as he can think of. but also is interested in a secindary notion of motion called vis or vis viva.
Before going on to that, it must be pointed out that Newton had to write his principia in a hurry. Thus these factal considerations are not highlighted here but in his speculations. He was a keen contributor to the corpuscular theory, and corresponded with Le Sage and Others on the nature of this force of inertia. What i have alluded to, the fractal nature of the second definition for the quantity of motion, like the first which is also fractal, both by a tautological device revealing the same, is much easier to spot in todays scientific status, since Benoit Mandelbrot drew attention to this essential relationship in space and nature. The significance of it is easily overlooked nd obscured by nnon complex thinkers who always simplify to mere nothing.
Potentially, tis fratal structure to a complex bodies quantity of motion explains the resultant motions of the orbits of the planets, without the need for gravitational force or curved space time as concepts, rather the uses of fractally applied angular and tangential quantities of motion.
The electromagnetic fields may provide the bais for this notion of quantity of motion applied fractally, and thus underpin the notions of curvature of space. In any case What Newton left out may be important to take his model forward on a fractal basis.
So we now turn to why Leibniz argued that the quantity of motion was to be described by the conjunvtion of the quantity of matter and the square of the velocity, as Huygens proposed and demonstrated experimentally. What quantity is the square of the velocity? one can only imagine it asan area enclosed by the velocity of an object as it moves through space in some great closed curve.