Hi Tigran and Steve. Thanks for the interest. I am not a physicist nor a mathematician so I will have to defer to you guys, but indulge my interest in the subject and allow me to be a devil's advocate for the sake of discussion.
The following is the current method of finding the derivative of the natural logarithm function.
dln(x)/dx = lim d→0 [ln(x+d)  ln(x)]/d =
lim ln[(x+d)/x]/ = lim(1/d) ln(1 + b/x) =
lim [ln(1 + d/x)^1/d]
Set u=d/x and substitute⎯
Lim u→0 [ ln (1 + u)^1/u ]
= 1/x ln [ lim u→0 (1 + u)^1/u ]
= 1/x ln(e) = 1/x
That derivative is false.
The step⎯ lim as u→0 [ ln (1 + u)^1/ux. ] = 1/x ln [ lim u→0 ( 1 + u)^1/u
Pulling the 1/x down from the exponent and putting it in front of the ln is permissible, but there was also a shifting of the lim u→forward, so that it is now in front of the ln, and this most certainly is not permissible. One cannot separate the ln from its number. The reason this is important is that because, as confirmation of the last step, we are sent to the definition of e⎯
e = lim n→0 (1 + n)^1/n
Take note that this does not read⎯
e = lim n→0 ln (1 + n)^1/n
These two equation are not the same and this invalidates not only the derivative of the natural logx , but invalidates the integral of 1/x as the nlx as Lebniz asserted.
The integral of 1/x according to the integrating method of Leibniz results in a contradiction; the undefined expression ‘1 divided by 0’.
∫1⁄x dx = x^1+1 / 1 + 1 = X^0 / 0 = 1/0
This is naturally unacceptable for mathematics to have this embarrassingly undefined expression. A so called ‘reasonable solution’ was accepted as the natural log x. The false derivation shown above was offered up as the solution. Because of the mathematical step of integrating the inverse function (1/x) is the most frequently used mathematical operation in physics (reflection), one can only shake one’s head at the sloppiness with which this science has ignored the natural logarithms. I came to the realization that motion is not calculable, if in fact, all motion is logarithmic.
Apollonius of Perga was a renowned Greek geometer most famous for his treatise of the conic sections we. Of interest in solving this problem of the impossible ellipse we can turn to another of his discoveries, the Apollonian circles. I cannot get a diagram of these circles on this forum but please Google it on Wiki for visual assistance.
These are two families of infinite circles such that every circle in the first family intersects every circle in the second family orthogonally. To better make the case, we must imagine this set of circle rotated 90 degrees and only then will it become obvious what we are revealing. The red circles all intersect at two points we will call F1 and F2. One type of polar coordinates is defined based on the Apollonian circles.
Algebraically, the circles in bipolar coordinate (σ, τ) is⎯
x = a [sinh τ ∕ cosh τ − cos σ]
y = a [sinh σ ∕ cosh τ − cos σ]
τ= ln (d1/d2) [When d1 = d2, τ is zero] I will refer to Wiki depiction of one of the infinite circles as d1 and d2 are the respective distance to any point on the circle from foci 1 and foci 2 (the poles of diagram).
Equivalently, the two equations above become⎯
x + iy = ai cot (σ + iτ / 2)
The take home lesson is that these circles are related to the hyperbolic functions and to the natural logarithm and, therefore to Euler’s number e, its base. The transcendentals π and Φ, as well as √1, the imaginary, are all included in this function. These functions are, therefore, not subject to the calculus. They are not derived from other functions nor can they have a defined integral.
These discoveries make the Apollonian circles clearly related to the family of infinite ellipses, our modified form of Euler’s Formula⎯
e^i∏ + Φ^2 + Φ = 0
It is time to take the circles of Apollonius and rotate them 90 degrees and see if we can visually identify this geometry with anything known in nature so that we can further reinforce its role in motion. Its shape is identical to a magnetic field of a rotating sphere.
You can see that the red circles of the Apollonian family look strikingly similar to the magnetic field lines of the Earth and the bar magnet. We have to wonder as to who really decided that these circles were due to a magnetic field and not a gravitational field. The poles of the Earth and the bar magnet correspond to the foci of circles in red. I think we have finally found the missing second focus in our ellipse!
I am going to suggest that the only field is the gravitational field and that these so called magnetic field lines are motions of charged particles controlled by the gravitational field. This new and improved gravitational field will indeed explain the infinitesimal changes in curvature (radius) of the elliptical orbit. What we have long held as a magnetic field was a gravitational field in disguised and in plain sight, somewhat like the Lone Ranger in his mask. The Sun has two foci from which its exerts its gravitational influence as do all other rotating orbs. The two foci form a gravitational field that is not spherical but conforms to the shapes of the Apollonian circles. This allows for a differential pull that controls the motion of the orbital. Gravity is thus greatest in the plane perpendicular to mass. This corresponds to the equatorial plane of the rotating spheroid. Mass can no longer be viewed as center and there is no center of gravity but a field generated by two foci of mass at the poles of our gyroscopic orb.
The orbital feels an ever changing pull of gravity at every point along the orbital path because the axes involved are fixed and the objects are oriented differentially as their axes are revolving in orbit. In this way, an orbital at aphelion can be pulled back as it experiences greater pull. At perihelion, a lesser gravitational force allows the orbital to escape its nearness to the Sun. The spheroidal field of Newton cannot do this because gravity cannot be allowed to fluctuate, so the inverse square law only compounded the problem of the orbital returning from aphelion and, conversely, crashing into the Sun at perihelion. My new theory of the gravitational field modeled geometrically by the Apollonian circles provides a very viable explanation of elliptical orbits and unifies the motion of the ellipse, welcoming it into the family of logarithmic spiral motion.
When we examine these field lines in light of my new theory of gravitation, we must conclude the these lines represent actual currents of charged particles moving from the southern hemisphere to the northern hemisphere. The Earth is a charged body and this charge creates potential. When there is an imbalance of motion across the equator, motion must seek balance with motion in the opposite direction. Because polarity exists, there is most definitely a differential in motion between hemispheres. More mass is in rotation below the equator than above it. Remember that the hemispheres are in counter rotation and this maintains the charge separation that creates polarity and potential. Because it is difficult for motion to balance across the reflective plane of the equator, particles of matter travel paths of least time in order to take motion (mass) away from the southern hemisphere and add it to the northern hemisphere. These electronic particles travel is these circumscribed paths showing the gravitational field in all of its glory. They even travel far out into so called empty space from the southern to the northern hemispheres. These particles, along with our atmosphere, form a protective shield to incoming solar winds and other harmful radiations of space which are shunted to the polar regions somewhat but mostly to open space. We recognize this as the polar lights and these particles interact in our ionosphere. Thus is the southern hemisphere slightly cooler than the northern hemisphere. Heat is the measure of electronic activity and electron showering down on the northern hemisphere has a heating effect. Hemispheric warming and cooling are occurring simultaneously and periodic warm ages and ice ages can be explained by precession (wobble) in the Earth’s axis and shifting of crustal and core mass in the giant gyroscopic dynamo we call home.
Newton flatly rejected the logarithmic spiral ellipse for reasons that had to do with his contention that the orbits would spiral into the Sun or out into space. Newton provided a rigorous proof in Principia that disallowed a logarithmic ellipse owing to his proof that an inverse cubic law would have to be in effect and this would not be inconsistent with EarthMoon observations. It is true that the poorly understood gravity attenuating properties of space provide the inverse square of distances between masses to roughly approximate the lessening in gravitational force. Newton provided a description of the elliptical orbit but failed to explained the forces involved in the motion.
42:11.5 Lineargravity response is a quantitative measure of nonspirit energy. All mass—organized energy—is subject to this grasp except as motion and mind act upon it. Linear gravity is the shortrange cohesive force of the macrocosmos somewhat as the forces of intraatomic cohesion are the shortrange forces of the microcosmos. Physical materialized energy, organized as socalled matter, cannot traverse space without affecting lineargravity response. Although such gravity response is directly proportional to mass, it is so modified by intervening space that the final result is no more than roughly approximated when expressed as inversely according to the square of the distance. Space eventually conquers linear gravitation because of the presence therein of the antigravity influences of numerous supermaterial forces which operate to neutralize gravity action and all responses thereto.
The differential axial tilts of the sun and planets in this gravitational field that acts preferentially in the plane perpendicular to the axes of rotation allows for infinitesimal changes in the radius of curvature of the elliptical path of the orbital.
Examine the catenary and see it as a visualization of gravity. The catenary is a logarithmic curve. Place a diagram of a true ellipse hanging on the wall with some tape. Get a fine linked chain and you see that you can fit the chain to the curvature of the chain. Wow!
Please let me know what you guys thing about all of this.
Regards, Louis
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