Proof 1

For my Kepler take, the initial mathematics consists of setting a circle [circumference to area] ratio equal to the ratio of any [circumference segment and its radially enclosed area]. So: [2pir/pir^2] is the definitive ratio for the efficiency of a circle. This reduces to [2/r]. This shows that the area enclosing efficiency of the circle, varies with [r].

Setting the definitive efficiency ratio equivalent to the ratio of any circumference segment of the circle, and its radially enclosed area, we have: [2pir/pir^2 = s/(rs/2)], where s is the length of the infinitessimal arc. This reduces to [2/r = 2/r], and shows that the definitve efficiency ratio is constant for any proportional part of a given circle. This aspect of the Euclidean circle is very simple, and rather mundane. It obviously follows from the spatial symmetry of the circle.

The above is a simple exercise in plane Euclidean geometry. 2nd year high school math. I suppose that before I tie this in to the orbital conics, I should explain why I consider the paragraphs above sufficient to describe a single C/S.

Historically, our study of this geometry implicitly included the assumption that it's space represented real world space. However, we did not define the Euclidean geometry space except in terms of area and volume and length. To enhance this study we constructed the regular orders of form from dimensionless points. Euclidean geometry, with it's uncoordinated space, allowed for the study of the pure mathematical ratios, wrt rigid form. Our implicit assumption that this space represented real world space, carried another assumption: Euclidean geometry forms could be constructed in, and applied to, every aspect of real world space.

The negative numbers and zero, together with the standard positive numbers were combined and then came Descartes (my chronological order here may be incorrect). Euclidean geometry and algebra were alloyed into the Cartesian frame. Algebra and geometry became analytical geometry. No further definition for Euclidean space was initially required because the assumption had always been that it represented real world space. So the continued implied assumption was that real world space was represented as Euclidean within the coordinated space. The regular static orders of form, made possible by Euclidean geometry space, were analyzed within the Cartesian C/S. Algebraic descriptions followed.

Galileo addressed human perception, wrt to the variance that would occur in the measurement of an event, from the perspective of observers in different states of motion. He derived an equation that corrected the variance in measure. The Galilean translational equation was thought to be sufficient at the time. During that time the only form of rapid transit was the ocean going vessel and the horse. So any algebraic equation describing the observed real world, could now be represented in what was regarded as 3D Euclidean (coordinated) space .

Physical experiments could be studied in this frame. These experiments included quantities like mass and velocity. Physical observations could be studied in this frame. These included Kepler's conics and their included, equal areas in equal times, and related ratios of time and space. Quantities that never existed in Euclidean geometry space led us to the vector. The Euclidean ratios were applicable to the coordinated vector space. The vector, a dynamic 2 dimensional geometric line further alloyed the static Euclidean space with observable real world space and strengthened our implicit assumption that real world space is Euclidean space.

Isaac Newton developed the calculus and the foundation for celestial mechanics using this C/S, by combining the work of Huygens, Descartes, Galileo, and Kepler. The assumption that Euclidean space described the real world continued until difficulties developed with electromagnetic space. Concurrently the Newtonian idea of instantaneous action at a distance wrt gravity, was noted and questioned. Einstein was born. He grew up as these ideas challenged the mainstream complacency. The means of rapid transportation then, was the railroad train. Einstein discovered as a bot, that he could appear to be moving while sitting in a non moving train. Einstein began questioning human perception.

You can see that the mainstream notion of Euclidean space did not rigorously define Euclidean geometry space according to its physical limitations. To gain a clear understanding of the properties attendant to a C/S, one must strictly define the extent to which its mathematics applies. Euclidean geometry space studies the ratios and relationships of rigid regular orders of form. Period.

I use that space as my first and most fundamental Cartesian C/S. In it the efficiency ratio of the circle is expressed as length and area. Now I wish to modify Euclidean geometry space to study an aspect of regular orders of form as they naturally occur, in real space. One manner they approximately occur, is as, trajectories of orbiting objects.

Kepler calculated three laws from the astronomical data left by Tycho Brahe. The first law states that the planet orbits are elliptical. Although we have since discovered that the orbits are not perfect ellipses, we can ignore that for the purpose of the present argument. I modify Euclidean geometry space in order to study the conic section in terms of a new property of space. Note that I am still not studying real space. I'm modifying the simple Euclidean geometry C/S in a most minimum manner to accommodate the new focus. Instead of static space, we have time-space. In niether case do we think these are representative of real space, beyond the limited application we are studying.

In our new space, the boundary of the circle represents a trajectory of an object, over time, and through space, only, if we view the passage of time as absolute, by setting the measure of that time, external to the frame. By definition the process we are measuring occurs within the space we have defined. We have added the property of motion to, and so modified Euclidean space. Motion requires time and space. Time-space is a property of, and internal to, our modified form of Euclidean geometry space.

Although a clock measures an interval of duration, a clock is still a clock. A clock is not time any more than a meter stick is space. They each measure a quantity which attends a process. During the time the classical mechanics was being developed and applied to the real world, the idea of time was considered as absolute. It was a subjective notion that rested on our sense of duration. So, the importance of placing time internal to the process itself had not been entertained. My inclusion of time as a part of the C/S, is not based on any idea of time dilation. It is solely based on accurately describing the reference frame we are using, within the context of the science at the time. I include it as a measurable property of the C/S, that must be taken into account, in terms of the physical ratio that defines it.

In this context, instead of [2pir/pir^2], we have the period [T/pir^2], as the first efficiency ratio. The second efficiency ratio is an interval [t] of the period [T], wrt any radially swept out area accompanying [t]. So we have: [T/pir^2] = [t/(rs/2)], where [t] is the infinitessimal time interval for any segment of the orbit, and [s] is the metric length of the infinitessimal arc, traveled during that time. This becomes [2t/rs = 2t/rs].

Now [2/r = 2/r], defined the proportionally symmetrical, consistent, efficiency property of the circle in static space. Where [2t/rs = 2t/rs] represents the approximate, equivalent, efficiency relationship in time-space. [2t/rs] is nearly constant for any given circular orbital trajectory, with a magnitude that varies, in direct proportion to twice the time and inversely proportional to the product [rs], wrt to different sized circle orbital trajectories. Note the ratio [t/s]. It is the ratio of time and distance and the inverse to velocity (here tangential). So that [t/s = 1/v]. Substituting, we have the simpler ratio: [2/rv] which is translationally equivalent to [2/r], wrt to the C/S they each apply to.

My conclusion here is that [2/rv] and [2/r] are translationally equivalent ratios that derive from the properties of a circle, from the perspective of two different C/S's.

This is the first section of the proof. It is supposed to show the C/S translational equivalence wrt the efficiency ratios [2/r] and [2/rv]. The next section of the proof will show that the efficiency property of equable areas in equal times, for any given orbital conic, is translationally equivalent (wrt each C/S) to the area time ratios, that would accompany a circle orbital trajectory, described by an orbiting object, at constant velocity.

II

In terms of efficiency, and wrt the two C/S's so far described, the orbital conic is equivalent to the Euclidean circle, in terms of the efficiency ratios: [2t/rs and 2/r]. To further this idea I will show that Kepler's law of areas is a logical consequence of the efficient geometric properties of the circle.