With circular motion there is a radius and circumference that may be measured as distance or duration. Call the spatial circumference *S*, and the temporal circumference *T*, which is known as the period. Distinguish the spatial and temporal versions of the radius, *R*, by using R and *Q.* Then *S* = 2π*R *and *T* = 2π*Q.* Also, *R *= *Qv*, and *Q* = *Ru*, with velocity, *v*, and legerity, *u*.

Circular orbits are a convenient entry into Kepler’s and Newton’s laws of planetary motion. Copernicus thought the orbits were circular, and most planetary orbits are in fact nearly circular. We have then three propositions *from the perspective of the Sun toward each orbiting planet*:

- Each planet orbits the Sun in a circular path with radius R.
- The Sun is at the center of mass of each planet’s orbit.
- The speed of each planet is constant.

Let’s follow the exposition given in *Elements of Newtonian Mechanics* by J.M. Knudsen and P.G. Hjorth (Springer, 1995), starting on page 6. Because the speed is constant, the acceleration follows the equation for uniform circular motion derived *previously*:

*a* = *v²* / *R*,

in which *v* is the speed and *a* is the acceleration. We have from the definition of speed:

*v* = *S* / *T* = 2π*R* / *T*.

Elimination of *v* from these equations leads to

*a* ∝ *R* / *T²*,

or *a* ∝ S / *T²*.

Kepler’s third law states that the orbital period, *T*, and the semi-major axis, *A*, are related as *T²* ∝ *A³*. For circular orbits this becomes

*T²* ∝ *R³*,

or *T²* ∝ *S³*.

Combining this with the equation for acceleration yields

*a* ∝ 1/*R²*,

or *a* ∝ 1/*S²*,

or *a* ∝ 1/*T ^{4/3}*,

or *a* ∝ 1/*Q ^{4/3}*.

Inserting the first acceleration into Newton’s second law leads to:

*F* ∝ *m* / *R²*,

with force, *F*, and mass, *m*. The force is directed toward the Sun, with a magnitude inversely proportional to the *square* of the distance from the Sun.

Because of Newton’s third law, there is an equal and opposite force from each planet toward the Sun. Which is to say:

*F* ∝ *M* / *R²*.

for mass, *M*. The combined law of gravitation is thus:

*F* = *GmM*/*R²*,

for some constant *G*.

Now consider gravity at the surface of the Earth. Newton showed that the gravitational force of a body would be the same if the mass were concentrated at its center, so the above equation can be used with the radius of the Earth, *ρ²*:

*F* = *GmM*/*ρ²* = *mg*,

with *g* as the acceleration of gravity on Earth. Then

*g* = *GM*/*ρ²*.

If the known values of *G*, *M*, and *ρ* are inserted into this equation, the result is *g* = 9.8 m/s².

Let us now modify this derivation so that distance is the independent variable instead of time. This is better called a law of *levitation* since it is naturally directed toward the smaller mass. We have then three propositions *from the perspective of the Earth, toward each transiting celestial body*:

- Each planet orbits the Sun in a circular path with period T.
- The Sun is at the center of vass of each planet’s orbit.
- The pace of each planet is constant.

Because the pace is constant, the expedition follows the equation for uniform circular motion derived *previously*:

*b* = *u²* / *Q*,

in which *u* is the pace and *b* is the expedition. Again, distinguish the spatial and temporal versions of the radius, *R*, by using *R* and Q. Then *S* = 2π*R* and *T* = 2π*Q*. Also, *R* = *Qv*, and *Q* = *Ru*, with velocity, *v*, and legerity, *u*.

We have from the definition of pace:

*u* = *T* / *S* = 2π*Q* / *S*.

Elimination of *u* from these equations leads to

*b* ∝ *Q* / *S²*,

or *b* ∝ *T* / *S²*,

or *b* ∝ *T* / *R²*.

Kepler’s third law states that the orbital period, *T*, and the semi-major axis, *A*, are related as *T²* ∝ *A³*. For circular orbits this becomes

*T²* ∝ *R³*,

or *T²* ∝ *S³*,

or *Q²* ∝ *S³*.

Combining this with the equation for expedition yields

*b* ∝ 1/*Q ^{1/3}*,

or *b* ∝ 1/*T ^{1/3}*,

or *b* ∝ 1/*S ^{1/2}*,

or *b* ∝ 1/*R ^{1/2}*.

Inserting the latter expedition into Newton’s second law in the form of *surge*, *Γ*, gives:

*Γ* ∝ *n* / *R ^{1/2}*,

with *surge*, *Γ*, and *vass*, *n*. The surge is directed away from the Sun, with a magnitude inversely proportional to the *square root* of the distance from the Sun.

Because of Newton’s third law, there is an equal and opposite *surge* toward each planet from the Sun. Which is to say:

*Γ* ∝ *N* / *R ^{1/2}*,

for vass, *N*. The combined law of *levitation* is thus:

*Γ* = *HnN*/*R ^{1/2},*

for some constant *H*. Since this is a function of the inverse square root instead of the inverse square, it is more sensitive to changes in *R*, i.e., distance above the surface of the Earth.

Consider levity at the surface of the Earth, using the above equation with the radius of the Earth, *ρ*:

*Γ* = *HnN*/*ρ ^{1/2}* =

*nh*,

with *h* as the expedition of levity on Earth. Then

*h* = *HN*/*ρ ^{1/2}*.

The values for *H*, *N*, and *ρ* may be inserted into this equation to determine the value of *h*.