---
title: "Inverse-square law"
chunk: 3/3
source: "https://en.wikipedia.org/wiki/Inverse-square_law"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T03:44:04.680534+00:00"
instance: "kb-cron"
---

== Non-Euclidean implications ==
The inverse-square law, fundamental in Euclidean spaces, also applies to non-Euclidean geometries, including hyperbolic space. The curvature present in these spaces alters physical laws, influencing a variety of fields such as cosmology, general relativity, and string theory.
John D. Barrow, in his 2020 paper "Non-Euclidean Newtonian Cosmology," expands on the behavior of force (F) and potential (Φ) within hyperbolic 3-space (H3). He explains that F and Φ obey the relationships F ∝ 1 / R² sinh²(r/R) and Φ ∝ coth(r/R), where R represents the curvature radius and r represents the distance from the focal point.
The concept of spatial dimensionality, first proposed by Immanuel Kant, remains a topic of debate concerning the inverse-square law. Dimitria Electra Gatzia and Rex D. Ramsier, in their 2021 paper, contend that the inverse-square law is more closely related to force distribution symmetry than to the dimensionality of space.
In the context of non-Euclidean geometries and general relativity, deviations from the inverse-square law do not arise from the law itself but rather from the assumption that the force between two bodies is instantaneous, which contradicts special relativity. General relativity reinterprets gravity as the curvature of spacetime, leading particles to move along geodesics in this curved spacetime.

== History ==
John Dumbleton of the 14th-century Oxford Calculators, was one of the first to express functional relationships in graphical form. He gave a proof of the mean speed theorem stating that "the latitude of a uniformly difform movement corresponds to the degree of the midpoint" and used this method to study the quantitative decrease in intensity of illumination in his Summa logicæ et philosophiæ naturalis (ca. 1349), stating that it was not linearly proportional to the distance, but was unable to expose the Inverse-square law.

In proposition 9 of Book 1 in his book Ad Vitellionem paralipomena, quibus astronomiae pars optica traditur (1604), the astronomer Johannes Kepler argued that the spreading of light from a point source obeys an inverse square law:

In 1645, in his book Astronomia Philolaica ..., the French astronomer Ismaël Bullialdus (1605–1694) refuted Johannes Kepler's suggestion that "gravity" weakens as the inverse of the distance; instead, Bullialdus argued, "gravity" weakens as the inverse square of the distance:

In England, the Anglican bishop Seth Ward (1617–1689) publicized the ideas of Bullialdus in his critique In Ismaelis Bullialdi astronomiae philolaicae fundamenta inquisitio brevis (1653) and publicized the planetary astronomy of Kepler in his book Astronomia geometrica (1656).
In 1663–1664, the English scientist Robert Hooke was writing his book Micrographia (1666) in which he discussed, among other things, the relation between the height of the atmosphere and the barometric pressure at the surface.  Since the atmosphere surrounds the Earth, which itself is a sphere, the volume of atmosphere bearing on any unit area of the Earth's surface is a truncated cone (which extends from the Earth's center to the vacuum of space; obviously only the section of the cone from the Earth's surface to space bears on the Earth's surface).  Although the volume of a cone is proportional to the cube of its height, Hooke argued that the air's pressure at the Earth's surface is instead proportional to the height of the atmosphere because gravity diminishes with altitude.  Although Hooke did not explicitly state so, the relation that he proposed would be true only if gravity decreases as the inverse square of the distance from the Earth's center.
Newton went up to independently develop and derive the inverse-square law for gravity in his Principia (1686). This later led to a priority debate between Newton and Hooke through correspondence.

== See also ==
Antenna (radio)
Distance decay
Fermi paradox
Flux
Gauss's law
Inverse proportionality
Kepler problem
Kepler's laws of planetary motion
Multiplicative inverse
Principle of similitude
Square–cube law
Telecommunications, particularly:
William Thomson, 1st Baron Kelvin
Power-aware routing protocols

== References ==
 This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 22 January 2022.

== External links ==
Damping of sound level with distance
Sound pressure p and the inverse distance law 1/r