G as a Metrological Artifact: A Formal Proof of Newton's Principle of Invariance

J. Rogers, SE Ohio

Abstract

This paper provides the mathematical formalism that underpins the historical thesis presented by J. Rogers in "The Constant of Proportionality." We demonstrate through a formal derivation that the gravitational constant, G, is not a fundamental constant of nature, but rather a composite scaling factor—a metrological artifact—that arises necessarily from the projection of an invariant, proportional physical law onto an arbitrary human system of units. By beginning with a dimensionless, "natural" form of the law of gravitation and performing a coordinate transformation into the SI system, we show that G emerges as the "Jacobian" of this transformation, containing all the necessary conversion factors. This formalism provides a rigorous proof that Isaac Newton's method of ratios was a mathematically sophisticated technique for performing calculations that were inherently independent of any such scaling factor, thereby validating the claim that his system was architecturally complete and unit-invariant.

1. Introduction: From Historical Thesis to Formal Proof

In a recent re-examination of Isaac Newton's work, Rogers (2025) argues that the absence of the gravitational constant G in the Principia Mathematica was not an omission, but a deliberate feature of a sophisticated methodology designed to be independent of arbitrary units. The thesis posits that Newton, working from a deep understanding of metrology, formulated his physics in the language of pure proportion to ensure that the unknown scaling constants would cancel out, a wisdom later lost as physics adopted an algebraic, unit-dependent framework.

While Rogers's argument is compelling on historical and philosophical grounds, it invites a rigorous mathematical demonstration. The purpose of this paper is to provide that demonstration. We will show that the modern algebraic form of Newton's law is a special case—a "projection"—of a more fundamental, invariant proportional law. In doing so, we will prove that G is not a property of gravity, but a property of our measurement system.

2. The Invariant Law in a Natural System

Let us begin by positing that the true physical law of gravity is a statement of pure proportionality, as Newton wrote it:

F ∝ m₁m₂/r²

This statement is dimensionless and unit-invariant. To work with it mathematically, we can define a "natural" system of units (such as Planck units) where the constant of proportionality is defined as unity. In this natural system, all physical quantities are expressed as dimensionless numbers (_nat). The law takes its simplest form:

Equation 1: The Invariant Law
F_nat = (m₁_nat ⋅ m₂_nat) / r_nat²

Here, m_nat is a pure number representing a quantity of mass in terms of a fundamental unit of mass, r_nat is a pure number for length, and F_nat is a pure number for force. This is the "business logic" of gravity, free from the conventions of any particular measurement framework.

3. The Projection into an Arbitrary Unit System (SI)

Humans, however, do not measure in natural units. We use arbitrary, conventional systems like the SI system (meters, kilograms, seconds). It is our local custom.  Any measurement we make in our lab (_SI) is related to its corresponding dimensionless "natural" value by a scaling factor. Let us denote the values of the natural units expressed in SI terms as m_P (the non reduced Planck mass in kg), l_P (the non reduced Planck length in meters), and F_P (the non reduced Planck force in Newtons).

The relationship between the systems is a simple scaling:

• m_SI = m_nat ⋅ m_P

• r_SI = r_nat ⋅ l_P

• F_SI = F_nat ⋅ F_P

To test the invariant law (Equation 1) using our lab measurements, we must first convert our SI data back into the dimensionless natural system by inverting these relations:

• m_nat = m_SI / m_P

• r_nat = r_SI / l_P

• F_nat = F_SI / F_P

Now, we substitute these expressions into the invariant law (Equation 1).  To keep the natural proportion the same we have to now cancel out the unit scaling we just added.  This tautology is explained in a later section:

(F_SI / F_P) = (m₁_SI / m_P) ⋅ (m₂_SI / m_P) / (r_SI / l_P)²

To yield a predictive formula for the force in our human units, we solve for F_SI:

F_SI = F_P ⋅ [ (m₁_SI ⋅ m₂_SI) / (m_P² ⋅ r_SI²) ] ⋅ l_P²

Rearranging this expression to group the measurement terms separately from the constant scaling factors, we get:

F_SI = [ F_P ⋅ l_P² / m_P² ] ⋅ (m₁_SI ⋅ m₂_SI) / r_SI²

We recognize this as the modern form of Newton's law, F = G ⋅ (m₁m₂/r²). This reveals the true identity of G:

Equation 2: The Definition of G
G ≡ F_P ⋅ l_P² / m_P²

This derivation demonstrates with mathematical certainty that G is not a fundamental constant. It is a composite artifact of conversion; a clump of scaling factors whose specific numerical value is entirely contingent on the arbitrary definitions of the meter, kilogram, and second.

4. The Tautology of Measurement: How G Encodes Complexity, Not Physics

At the heart of our derivation is a step that appears, on the surface, to be a meaningless mathematical tautology. When we substitute our normalized measurements into the invariant law, we are implicitly working with a structure like:

F_nat = (F_nat * F_P) / F_P

Mathematically, this is trivial (x = (x*y)/y). Physically, however, it is a profound statement about the process of measurement. This operation reveals the true function of constants like G: they do not add new physical information, but rather encode the complexity of our chosen measurement framework.

Let's break down the physical meaning of the three parts of this "tautology":

1. F_nat: This is the pure, dimensionless result of the physical law. It is the "truth" of the physics, independent of any observer's tools.

2. F_nat * F_P: This is the act of scaling. We take the pure number from the law and multiply it by a physical scaling factor (F_P, the Planck Force in Newtons) to produce a quantity that can be measured on a human instrument. The result is F_SI, a physical quantity with units.

3. ( ... ) / F_P: This is the act of normalizing. We take the human-measured quantity (F_SI) and divide it by the same physical scaling factor to strip the units away and return to the pure, dimensionless number (F_nat) that the invariant law requires.

The entire algebraic structure of modern physics, with its explicit constants, is built upon this two-step process of scaling and re-normalizing. The constant G is not a discovery; it is the pre-packaged result of this round trip. It contains all the scaling information (F_P, l_P, m_P) required to translate our arbitrary SI measurements into the "natural" language the universe understands, and then back again.

Therefore, the introduction of G into the formula adds zero new physical information about gravity. All it does is encode the arbitrary conventions of the SI system directly into the law. It adds mathematical complexity for the sole purpose of compensating for the complexity we introduced by choosing our own rulers in the first place.

5. G as a Coordinate System Jacobian

The role of G is best understood by analogy to the Jacobian determinant in multivariable calculus. When one transforms a function from one coordinate system to another (e.g., Cartesian to polar), the Jacobian determinant appears as a scaling factor that accounts for how the geometry of space is stretched or compressed by the new coordinate basis.

In exactly the same way, G is the "metrological Jacobian" for the transformation from nature's invariant coordinate system to our arbitrary human one. It is the scaling factor required to ensure the law remains valid after we have "distorted" it with our idiosyncratic choice of units. It contains no new physical information about gravity; it contains only information about our rulers and scales.

6. Mathematical Vindication of Newton's Method of Ratios

This formalism provides a powerful lens through which to re-evaluate Newton's methodology. His consistent use of ratios was not a primitive workaround but a sophisticated mathematical technique to avoid the tautological complexity described above.

Consider his Moon-Apple test. Newton compared the acceleration of the apple (a_apple) to that of the Moon (a_moon). In our formalism:

a_apple = F_apple / m_apple = G ⋅ (M_earth ⋅ m_apple) / (R_e² ⋅ m_apple) = G ⋅ M_earth / R_e²
a_moon = G ⋅ M_earth / r_m²

When Newton takes the ratio, he is performing the following calculation:

a_apple / a_moon = (G ⋅ M_earth / R_e²) / (G ⋅ M_earth / r_m²)

The composite scaling factor G (the "Jacobian") and the unknown mass of the Earth M_earth (another scaling factor in this context) cancel out perfectly, leaving a pure, dimensionless geometric statement:

a_apple / a_moon = r_m² / R_e²

Newton's method allowed him to perform the calculation as if he were working directly with the natural, dimensionless quantities, bypassing the need to ever compute the value of G. He worked in a system where the tautology of (x*y)/y was unnecessary because y—the scaling factor for any given unit system—was designed to be cancelled from the outset.

7. Conclusion

The mathematical evidence is unequivocal. The gravitational constant G is not a fundamental parameter of the universe. It is a composite conversion factor that emerges from the act of measurement itself—a necessary artifact when an invariant proportional law is expressed in an arbitrary, human-defined unit system.

This conclusion provides the rigorous mathematical support for the historical thesis proposed by Rogers. Isaac Newton was not waiting for a missing constant. He architected his entire system in the Principia to be independent of such metrological contingencies. His method of ratios was a deliberate and brilliant technique for canceling out the "Jacobian" of measurement, allowing him to prove the universality of his law without reference to any particular set of units. The modern interpretation has mistaken the calibration constant of our tools for a deep property of nature, a category error that Newton's original, more abstract methodology wisely avoided.