The Ratio of Speeds

I guess Chris Huygens has a deserved place as the father of time.

We actually have to go back to Descartes and his philosophy to get a sense of Chris impact.
Descartes drew philosophically on his jesuit upbringing in formulating through his meditative praxis a consistent philosophy of existence under god. He had noqualms at starting with god in some universal notion derived from centuries of theological philosophy and greek style deduction. Where eastern culture clashed with greek the test was trial by contest. This is the underlying notion of" logical" proof. Whichever individual representing a "conviction of truth"who won the often bloody trial was vindicated as by divine sanction. Today we carry the notion by logically or rather systematically consistent propositions winning the day over competing proportions when empirically tested.

So called truth if subject to empirical test is only as good as its last vindication.

However logicians, magi and magicians have long been aware of and utilised the tautology. By denying inexperienced "souls" the full knowledge of tautology, and encouraging hypnogogy in the same magicians were able to define the experience of reality pretty much how they wished, and to sparingly utilise naturally occurring consequents of acceptance and attentional focus. We really do "see the world through rose tiinted glasses".

So Descartes brilliantly organized on the basis of "pure reason" that is "Logos" as a type of gods wisdom and thinking action,personified in the Christ and the spirit, his take on the structure of the universe under god.
Of course it is a fusion of cultural thinking traceable back to the Sumerians, but studiously filtering out any notions that did not support his theology or christology, simply denoting such dissonance as error, evil, sin, pagan etc.

Nevertheless despite this bias Descartes philosophy is remarkably similar to philosophies found at this level of erudition in all cultures .

Therefore Leibniz, and other philosophers of reason and empiricism including Barrow, Wallis and Newton, were given an established idea to work against. It is to be noted that no one essentially dismissed Descartes, they merel wanted to improve upon his notions.it is thereby to be noted that unless one specifically subscribes to a cultural philosophy other than western the basic presiding philosophy is Cartesian in all of science.
Kant later challenges Cartesian philosophy with a variant devised by Newton, and Newton's variant has only just recently been challenged.

Leibniz, drawing on empirical evidence from Galileo and Huygens and other mechanics argued that descartes notions of vis were flawed by tautology. That to clear things in a more consistent way one needed to view motion as "independent" of distance, and that the god preserved and conserved "quantity" in bodily interactions was not matter times by the velocity or rather speed of that matter but rather matter times by the speed squared .

This rather technical argument takes some reviewing of apprehensions. Until Galileo there was no real distinction beyond matter moving at variable speeds. Speeds if they were measured at all, were taken as axiomatic attributes of moving matter. Some experience was allowed to inform the opinion of the different amounts of matter and the different speeds and the interactive impact of these apprehensions in collisions.

It was commonly deduced that heavier items travel slower along the ground than lighter ones, but fall faster!

After common thought speed was understood but measured differently in different circumstance: sometimes it would be measured by distance traveled in a day, other times by days taken to travel a distance between known landmarks. Distance was therefore the underlying measure of speed. A greater speed meant a greater distance covered, a variation in the speed meant a variation in the distance covered.

So 2 examples of invariance were ver puzzling. The first was Gallileos invariance in speed with regard to mass: heavier objects fall at the same speed as relatively lighter ones. Hang on a minute, the objects do not show a uniform speed so Gallileo concluded that the objects varied their speed in exactly the same way and he wanted to demonstrate this surprising result by proportions. Therefore he measured all sorts of proportions including a notion he called musical time or rhythm. He showed that for constant musical time the proportions of distance achieved were identical. The rhythm of motion , the music of the spheres has its origins ib Galileo's methods.

At the same time Huygens and others were investigating the motion of a pendulum. The invariance here is no matter what the initial pressure on the weight the periodic motion seemed to adjust to give the same rhythm. So clearly the farther the pendulum had to fall the faster its speed at the bottom of the swing, but the rhythm of the music of the spheres seemed to remain constant.

Chris was able again by taking all sorts of measurements to show a proportion between two pendulums which related the length of the pendulums radius to the rhythm of its period, and the relation was a proportion to the square root of the radius .

The common name for this rhythm was "time". Thus "time" if anything is an analogy of musical time that is a constant driving rhythm which of course has a direction associated with it from the spatial representation on music paper. Our confused notion of time has its origin in music, and both invariances are crucial to music making. The gravitational pressure on a mass drives the pendulum of a metronome that beats out time in exactly the proportions Huygens observed. These proportions have their resolution in the Euclidean geometry of the circle.

As a consequence of Chris's work Leibniz felt that the conserved quantities would have to depend on mass times by the square of the speeds/velocity, and all by default used definitions of motion dependent on the notion of musical timing. Musical timing only became scientific timing when Huygens introduced the first accurate clocks based on the invariant properties of falling matter and pendulum action. However it should not escape notice that scientific time is based on motion and a comparison of motions is what rhythmical time is. The speed of the contiguous parts of the pendulum vary but the rhythm we apprehend as constant: thus constant speed analogues of Huygens pendulum clocks can and have been made which further highlight the comparisons of speed or motion denoted as time.
It is also important to note that periodic motion follows a closed motion trace which can be measured as a distance, and the motion trace in 3d space can be projected onto 2d space to produce a trace. Finally if that 2d space is in relative motion to the anchor point of the pendulum bounded traces can be described which analogise the comparison of a motion against a motion, a distance against a distance, and a rhythm against a rhythm.

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