Newton ’ s Parabola Observed from Pappus ’ Directrix , Apollonius ’ Pedal Curve ( Line ) , Newton ’ s Evolute , Leibniz ’ s Subtangent and Subnormal , Castillon ’ s Cardioid , and Ptolemy ’ s Circle ( Hodograph ) ( 09 . 02 . 2019 )

Johannes Kepler and Isaac Newton inspired generations of researchers to study properties of elliptic, hyperbolic, and parabolic paths of planets and other astronomical objects orbiting around the Sun. The books of these two Old Masters “Astronomia Nova” and “Principia...” were originally written in the geometrical language. However, the following generations of researchers translated the geometrical language of these Old Masters into the infinitesimal calculus independently discovered by Newton and Leibniz. In our attempt we will try to return back to the original geometrical language and to present several figures with possible hidden properties of parabolic orbits. For the description of events on parabolic orbits we will employ the interplay of the directrix of parabola discovered by Pappus of Alexandria, the pedal curve with the pedal point in the focus discovered by Apollonius of Perga (The Great Geometer), and the focus occupied by our Sun discovered in several stages by Aristarchus, Copernicus, Kepler and Isaac Newton (The Great Mathematician). We will study properties of this PAN Parabola with the aim to extract some hidden parameters behind that visible parabolic orbit in the Aristotelian World. In the Plato’s Realm some curves carrying hidden information might be waiting for our research. One such curve the evolute of parabola discovered Newton behind his famous gravitational law. We have used the Castillon’s cardioid as the curve describing the tangent velocity of objects on the parabolic orbit. In the PAN Parabola we have newly used six parameters introduced by Gottfried Wilhelm Leibniz abscissa, ordinate, length of tangent, subtangent, length of normal, and subnormal. We have obtained formulae both for the tangent and normal velocities for objects on the parabolic orbit. We have also obtained the moment of tangent momentum and the moment of normal momentum. Both moments are constant on the whole parabolic orbit and that is why we should not observe the precession of parabolic orbit. We have discovered the Ptolemy’s Circle with the diameter a (distance between the vertex of parabola and its focus) where we see both the tangent and normal velocities of orbiting objects. In this case the Ptolemy’s Circle plays a role of the hodograph rotating on the parabolic orbit without sliding. In the Plato’s Realm some other curves might be hidden and have been waiting for our future research. Have we found the Arriadne’s Thread leading out of the Labyrinth or are we still lost in the Labyrinth?


Introduction
The famous quote of Heraclitus "Nature loves to hide" was described in details by Pierre Hadot in 2008.Hadot in his valuable book gives us many examples how Nature protects Her Secrets.In several situations the enormous research of many generations is strongly needed before the right "recipe" unlocking the true reality can be found.Conic sections -Circle, Ellipse, Hyperbola, and Parabola -are among the oldest curves, and belong to the Treasure of Geometry.The conic sections seem to have been discovered by Menaechmus and were thoroughly studied by Apollonius of Perga (The Great Geometer) and his scholars.The conics are endowed with numerous beautiful properties, some those properties are shared by the entire family, while some properties are unique and original for each of them.In our research we have to be very careful and patient when we want to apply those properties for the planet orbits.Menaechmus said to Alexander the Great: "O King, for travelling over the country, there are royal roads and roads for common citizens, but in geometry there is one road for all." Parabola is a very original conic section with its own Beauty and Secrets.Though, it has only one focus, it might reveal similar properties as Her Sisters Ellipse and Hyperbola.Pappus of Alexandria discovered the directrix and focus of the parabola, and Apollonius of Perga systematically revealed numerous properties of the parabola.This Ancient Treasure passed into the hands of Copernicus, Galileo, Kepler, Huygens, Newton, Leibniz and many others.
W.R. Hamilton in 1847 discovered how to find the tangent velocity for the elliptic orbit using the auxiliary circle of that ellipse.This technique works very well for hyperbola with two foci, too.But how to find the tangent and normal velocities for parabola with only one focus?
The support came to us from the Great Old Master -Gottfried Wilhelm Leibniz.Gottfried Wilhelm Leibniz during his preparation work on the infinitesimal calculus introduced six parameters for a point on the parabola: abscissa, ordinate, length of tangent, subtangent, length of normal, and subnormal.For his infinitesimal triangle Leibniz used the ratio of ordinate to subtangent and the ratio of subnormal to ordinate.Another important inspiration came to us from Johann Castillon with his cardioid -the inversion curve to parabola.This curve might bring an information about the tangent velocity of an object on parabolic orbits.
Can we try to employ the Castillon -Leibniz concept in order to get the needed tangent and normal velocities for objects on parabolic orbit?Can we find a hodograph circle around the PAN Parabola?Who can support us in this research?How about the Great Old Master -Claudius Ptolemy and his Circles?
A possible chance for further classical development of the Newton's parabola is to penetrate more deeply into the secrets of the Newton's parabola and to reappear with some new hidden properties overlooked by earlier generations of researchers.Our guiding principle is the existence of the Plato's Realm with invisible mathematical objects that might bring to us some additional information about the visible Newton's parabola in the Aristotelian World.In this contribution we have been working with these mathematical objects from the Plato's Realm: 1. Parabola properties discovered by Apollonius of Perga -the Great Geometer -and many his scholars.4. The interplay of the directrix of parabola P (= Pappus -the discoverer of directrix), the vertex of the parabola A (= Apollonius of Perga -the Great Geometer), and the occupied focus N (= Isaac Newton -the Great Mathematician) together forms the PAN Parabola with several interesting hidden properties of those parabolic paths.
5. Castillon's cardioid (the inverse curve to parabola) as the measure of tangent velocities on parabolic orbits.6. Pedal curve with the pedal point in the Newton's focus -the "auxiliary circle" -the line perpendicular to the axis of parabola in the vertex.
7. Tangent and normal to a point on the parabolic orbit described by six parameters introduced by Gottfried Wilhelm Leibniz: abscissa, ordinate, length of tangent, subtangent, length of normal, and subnormal.8. Moment of tangent momentum, moment of normal momentum.9. Hodograph: Ptolemy's Circle with the diameter a (distance between the vertex of parabola and its focus) where we see both the tangent and normal velocities of orbiting object.In this case the Ptolemy's Circle plays a role of the hodograph rotating on the parabolic orbit without sliding.
The experimental analysis of properties of the PAN Parabola should reveal if we have found the Arriadne's Thread leading out of the Labyrinth or if we are still lost in the Labyrinth.
(We are aware of the famous quote of Richard Feynman from the year 1962: "There's certain irrationality to any work in gravitation, so it is hard to explain why you do any of it.")

Some Properties of the PAN Parabola
Figure 1, Figure 2, and Figure 3 show some parabolic properties of the PAN parabola that might be used for the description of motion of objects around the Sun on the parabolic orbits.Some of those parameters are very well-known from many good books on conic sections, some parameters are newly derived.The great inspiration was found in the works of Isaac Todhunter (1881), who studied very deeply the properties of conic sections.

Proposed Reflecting and Refracting Properties of Solar and Object Gravitons on the Parabolic Orbit
In this section we will assume that Solar gravitons enter into the internal volume of objects on the parabolic orbits and collide with object gravitons in four possible scenarios as it was in details described by Jiří Stávek (2018).
Gravitons are reflected and refracted similarly as in the case of the Kepler's ellipse, and Newton's Hyperbola.Therefore, we will present here only a qualitative schema of reflective and refractive properties of parabola in Figure 4. We have found this remarkable formula for the tangent velocity v T within several minutes: ( where v T is the tangent velocity, v C is the circular velocity, φ is the angle between the axis of parabola and the line connecting the Newton's focus with the object on parabolic orbit, v ESC is the very well-known escape velocity for the parabolic orbit. The great inspiration for this concept was delivered to us by the book of Viktor Blåsjö (2017) "Transcendental Curves in the Leibnizian Calculus" where author describes in details the works of Old Masters before the infinitesimal calculus started to dominate the book "Principia…".

Newton's Parabola Observed from the Newton's Evolute
Newton discovered several important properties hidden in the Kepler's ellipse, the Newton's Hyperbola and the Newton's Parabola in his Principia in 1687.For the centripetal force F, he derived formula: ( 2 ) where m is the mass of the planet (or any object), v T is the tangent velocity of the planet and ρ is the radius of curvature of that ellipse, hyperbola or parabola.The locus of radii of curvature is termed as the evolute.This equation opened completely new possibilities in the understanding of the Kepler's ellipse, Newton's Hyperbola, and Newton's Parabola.We will use this procedure for the description of events in Newton's parabola.
In the standard procedure both quantities v T and ρ are found by the infinitesimal calculus discovered independently by Newton and Leibniz.
We will present here the trigonometric approach to these two quantities (v T in the next chapter).The radius of curvature of the hyperbola ρ can be derived in the trigonometric way shown in Figure 5. Figure 5 describes an interplay between the normal to the tangent and the line connecting the Sun and orbiting planet.
( ) . Trigonometric approach to reveal the expression for the radius of curvature ρ of the Newton's parabola We have used the deep knowledge of parabola properties of Issac Todhunter (1881) and extracted from Figure 5 the expression for the radius of curvature, which is already known in the existing literature: ( 3 ) (The quantities expressed in the trigonometric language are simpler and Nature can talk with us in this trigonometric language that could be depicted in simple Figures without words.The trigonometric function cosα is cleverly hidden in the Equation 3).

Newton's Parabola Observed from the Pedal Curve with Pedal Point in the Newton's Focus and from the Leibniz's Subnormal
The pedal curve of the Newton's parabola is the locus of the feet of the perpendiculars from the occupied focus to the tangent of that parabola.In this case the pedal curve is the famous "auxiliary circle" of the parabola -the line perpendicular to the axis of parabola in the vertex of that parabola.The distance JN can be used for the arm of the moment of tangent momentum -see Figure 6.We were inspired by W.R. Hamilton who in 1847 discovered his concept for the Kepler's ellipse that is known as the hodograph.This approach was several times forgotten and its Beauty was several times rediscovered by many researchers.E.g., Richard Feynman in his "Lost lecture" made this concept very well known for our generation.However, this great idea does not work for parabola -there is no empty focus of parabola.
( ) Luckily, we were inspired by another Great Old Master -Gottfried Wilhelm Leibniz.Gottfried Wilhelm Leibniz during his preparation work on the infinitesimal calculus introduced six parameters for a point on the parabola: abscissa, ordinate, length of tangent, subtangent, length of normal, and subnormal.Leibniz studied the change of these six parameters and their dependence on the direction of tangent and normal and their dependence on the curvature in that point on the parabola.For his infinitesimal triangle Leibniz used the ratio of ordinate to subtangent and the ratio of subnormal to ordinate.Can we extract any additional information hidden in those six Leibniz's parameters?For details see D. Dennis (1995), D. Dennis and J. Confrey (1995).[See also A. R. Hall (2015) and T. Sonar ( 2018)].
For the tangent velocity v T of an object on the parabolic orbit we propose to use the ratio of the subnormal to the length of the normal: where v ESC is the very well-known escape velocity just to leave the Sun or the planet.The first part of this Equation was given to us by Castillon from his cardioid, the second part of that Equation was presented to us by G.W. Leibniz.Now, we can test the validity of the Newton's formula expressed in the trigonometric language and compare these trigonometric formulae with formulae obtained in other mathematical languages.
The famous Newton's formula can be trigonometrically expressed as: ( 5 ) We have inserted the expression for the escape velocity v ESC = 2 ½ v C , where v C is the velocity for the circular orbit.At the end we have obtained the standard gravitational parameter µ: ( 6 ) where a is the parameter of the PAN Parabola and v C is the circular orbital velocity for that object.  2 For the normal velocity v N of an object on the parabolic orbit we propose to use the ratio of the subtangent to the length of the tangent: ( 7 ) where v ESC is the very well-known escape velocity of an object just to leave the Sun or the planet.The moment of momentum L is defined as the product of the linear momentum with the length of the moment arm, a line dropped perpendicularly from the origin onto the path of the particle.It is this definition: L = (length of moment arm) x (linear momentum).
The moment of the tangent momentum L T for the Newton's parabola is given as: ( 8 ) where m is the mass of an object, v T the tangent velocity of an object on the parabolic orbit planet and JN is the length of the moment arm (the distance between the Newton's occupied focus and the tangent to the point F).The moment of the tangent momentum L T is constant during the complete parabolic path of the Newton's parabola.Therefore, there is no contribution to the torque from this moment of the tangent momentum.This is very well-known experimental fact documented in the existing literature.
The moment of the normal momentum L N for the Newton's parabola is given as: ( 9 ) where m is the mass of an object, v N the normal velocity of an object on the parabolic orbit and NC is the length of the moment arm (the distance between the Newton's occupied focus and the tangent to the point C).The moment of the normal momentum is constant during the complete path of the Newton's parabola.Therefore, there is no contribution to the torque from this moment of the normal momentum.This is very well-known experimental fact documented in the existing literature.

"Antikythera Mechanism" in the Solar System
We propose to use the very-well known Antikythera Mechanism as an analogy for the visible PAN Parabola -a part of our Aristotelian World -connected deeply with invisible curves from the Plato's Realm -Pappus' Directrix, Apollonius' auxiliary Circle, Newton's Evolute, Castillon's cardioid, Leibniz's six parameters (abscissa, ordinate, length of tangent, subtangent, length of normal, and subnormal), Ptolemy's Circle (Hodograph).
Are there some more hidden curves in the Plato's Realm connected to the PAN Parabola?How to distinguish the real physical meaning written in those curves from fictious events if both are mathematically correct?The detailed analysis of these ideas we will leave to the Readers of this Journal better educated in mathematics.
5. We have employed Leibniz's length of the tangent and subtangent to get an expression for the normal velocity of an object on the parabolic orbit.

2 .
Directrix and focus of parabola discovered by Pappus of Alexandria 3. Locus of the radii of curvature (evolute) of parabola -Isaac Newton in 1687.

Figure 1 .Figure 2 .Figure 3 .
Figure 1.Some properties of the PAN Parabola: Directrix P (=Pappus), vertex A (=Apollonius), occupied focus N (=Newton), tangent, normal, focal chord Some properties of the PAN parabola y 2 = 4 ax a = 1 = PA = AN … parameter of the PAN parabola Latus rectum = 4a PANTHEION' Circle for the construction of the tangent and the normal to a point F (Πάνθειον) P … Pappus' directrix A … Apollonius' vertex N … Newton's occupied focus F … object on the parabolic orbit J, S … points on the tangent ρ … radius of curvature of hyperbola Leibniz's six parameters of a point on the parabolic orbit

Figure 4 .
Figure 4. Reflective and refractive properties of parabola

Figure 6 .
Figure 6.PAN Parabola with the distance JN (length of the arm for the moment of tangent momentum) and with the SURF Circle (Ptolemy's Circle -the Hodograph) where SU represents the tangent velocity on the parabolic orbit 7. Newton's Parabola Observed from the Contrapedal Curve with Pedal Point in the Newton's Focus and from the Leibniz's SubtangentWe want to find an expression for the normal velocity v N of an object on the parabolic orbit.In this case the PAN Parabola shows Her original Feature and Beauty.We will use the contrapedal distance NC for the arm of the moment of the normal momentum from Figure7.

Figure 7 .
Figure 7. PAN Parabola with the distance NC (length of the arm for the moment of normal momentum) and with the SURF Circle (Ptolemy's Circle -the Hodograph) where SF represents the normal velocity on the parabolic orbit

8.
Moment of the Tangent Momentum and the Moment of the Normal Momentum of the Newton's Parabola Based on the formulae in Tables I and in Figures in this contribution we can evaluate the moment of the tangent momentum L T and to introduce a new physical quantity -the moment of the normal momentum L N .

Table 1
summarizes some relations for the PAN parabola.

Table 1 .
Some properties of the PAN parabola