Johannes Kepler and Inverse-Square Laws

In my post, Two Productive Research Programs in the Early 19th Century, one of the programs that I wrote about was the assumption of action-at-a-distance. I presented Isaac Newton’s law of universal gravitation (1687), Charles-Augustin de Coulomb’s electrical (1785) and magnetic laws, André-Marie Ampère’s force law between electrical current elements (1822) and Wilhelm Eduard Weber’s force law between charged particles (1846). All of these laws are examples of inverse-square laws, which state that the intensity of the instantaneous mutual attraction or repulsion along a straight line between two entities, varies inversely with the square of the distance between these two entities.

The idea of an inverse-square law follows directly from the fact that the surface area of a sphere of radius r is 4π r². If the radius r of a sphere is, say, doubled, the surface area of that sphere will be quadrupled. So if we view attraction or repulsion as acting equally in all directions from the center of a sphere, then the intensity of that attraction or repulsion at a point on the surface of the sphere will lessen as the radius increases. Given that the surface area of the sphere varies with the square of the radius, it is natural to assume that the intensity of the attraction or repulsion at a particular point on the sphere will vary inversely with the square of the radius of the sphere.

The first inverse-square law was presented by Johannes Kepler in 1604, with respect to the intensity of light from a given source, in his Astronomiae pars optica¹ [The Optical Part of Astronomy]:

PROPOSITIO IX

Sicut se habent sphaericae superficies, quibus origo lucis pro centro est, amplior ad angustiorem: ita se habet fortitudo seu densitas lucis radiorum in angustiori, ad illam in laxiori sphaerica superficie, hoc est, conuersim. Nam per 6. 7. tantundem lucis est in angustiori sphaerica superficie, quantum in fusiore, tantò ergo illic stipatior et densior quàm hîc. Si autem radii linearis alia atque alia esset densitas, pro situ ad centrum (quod Prop. 7. negatum est) res aliter se haberet. [p.22, my emphasis]

Here is an English-language translation with the help of Google Translate:

PROPOSITION IX

Just as spherical surfaces, in which the source of light is at the center, are larger to narrower: so the strength or density of the rays of light in a narrower spherical surface is similar to that in a looser spherical surface, that is, inversely. For by 6.7. there is as much light in the narrower spherical surface as in the more fused one, therefore it is as much denser and denser there than here. If, however, the linear radius were of different density, according to its position at the center (which is denied in Prop. 7), the situation would be different. [my emphasis]

What is interesting is that in his previous work, Mysterium Cosmographicum² (1596), Kepler wrote the following:

As it is, however, this ratio of the motions is compounded with the weakness of the moving spirit in the more distant planet. Therefore we must also discover what its relationship is with this weakness. Let us suppose, then, as is highly probable, that motion is dispensed by the Sun in the same proportion as light. Now the ratio in which light spreading out from a center is weakened is stated by the opticians. For the amount of light in a small circle is the same as the amount of light or of the solar rays in the great one. Hence, as it is more concentrated in the small circle, and more thinly spread in the great one, the measure of this thinning out must be sought in the actual ratio of the circles, both for light and for the moving power. [p.201, my emphasis]

The commentary notes for the emphasized sentence read as follows:

At this time Kepler evidently believed that the intensity of light weakened in proportion to distance from the source (he speaks of light spreading out in a circle, not a sphere) and he concluded that the effect of the moving soul in the sun weakened in this way. When he discovered the inverse-square law for the intensity of light (Astronomiae pars optica, chapter 1, prop.9, KGW 2, p.22), Kepler was able to retain the inverse-distance law for the moving force in the sun, because a force spreading out in the plane of the ecliptic sufficed to explain the motion of the planets. [p.249, n.6]

So in 1596, Kepler wrote that “Let us suppose, then, as is highly probable, that motion is dispensed by the Sun in the same proportion as light,” at the time supposing that these both weakened in proportion to distance from the source. However, in 1604, Kepler had determined that the intensity of light weakens in proportion to the square of the distance from the source. But he did not update his understanding of the motion “dispensed by the Sun.”

What a missed opportunity!

Kepler continued his research in astronomy, publishing his first two laws in the Astronomia Nova (1609) and the Third Law in the Harmonices Mundi (1619).

What is fascinating is that the Third Law is implicitly inverse-square. In fact, if one adds the famous F = ma (force = mass × acceleration) from Newton, only trivial manipulations are needed to pass from Newton’s Law of universal gravitation to Kepler’s Third Law, and vice versa.

So it would not be Kepler, but Ismaël Boulliau (Ismaël Bullialdus) who would be the first to write in 1645 in his Astronomia Philolaica³ about the inverse-square relationship for what we now call gravity:

Virtus autem illa, qua Sol prehendit seu harpagat planetas, corporalis quae ipsi pro manibus est, lineis rectis in omnen mundi amplitudinem emissa quasi species solis cum illius corpore rotatur: cum ergo sit corporalis imminuitur, & extenuatur in maiori spatio & interuallo, ratio autem huius imminutionis eadem est, ac luminis, in ratione nempe dupla interuallorum, sed euersa. Hoc non negauit Keplerus, attamen virtutem motricem in simpla tantum ratione inueruallorum contendit imminui: ait insuper illam virtutis attenuationem solummodo proficere ad debilitandam virtutem motricem causa longitudinis, quia motus localis, quem Sol planetis insert, tantum sit in longitudinem, in quam etiam ipsius Solis partes corporis sunt mobiles, non etiam in latitudinem: compensat vero defectum huius analogiae multiplicando materiam planetae tardioris. [p.23, my emphasis]

Here is an English-language translation with the help of Google Translate:

Now that power, by which the sun grasps or grasps the planets, is corporeal, which is for its own hands, sent forth in straight lines into the whole extent of the world, as if the species of the sun revolves with its body, the diminution is the same as that of light, namely, in the ratio of twice the intervals, but it is reversed. Kepler did not deny this, but he contends that the motive power is diminished only by the simple reason of the facts: he says, moreover, that the attenuation of the power only proceeds to weaken the motive power on account of longitude, because the local motion which the Sun puts into the planets is only in the longitude, in which also the parts of the Sun itself they are mobile in body, not even in latitude: indeed he compensates for the lack of this analogy by multiplying the matter of the slower planet. [my emphasis]

What Boulliaud is writing here that “Kepler did not deny this,” i.e., that Kepler was so close to figuring this out. However, Boulliaud continues, Kepler did not accept these conclusions, assuming that gravity only applied in two dimensions, i.e., on the ecliptic plane, and not three.

The irony is that Boulliaud himself did not accept Kepler’s first two laws, believing that the orbits of the planets were circular, not elliptical.

The clarification of all of these issues would only come with the publication of Newton’s Philosophiæ Naturalis Principia Mathematica (1687).

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1. Johannes Kepler. Astronomiae pars optica (1604). In Johannes Kepler, Gesammelte Werke, Band II. Herausgegeben von Franz Hammer. München: C.H.Beck’sche Verlagsbuchhandlung, 1939.

2. Johannes Kepler. Mysterium Cosmographicum: The Secret of the Universe. Translation by A.M. Duncan. Introduction and Commentary by E.J. Aiton. With a Preface by I. Bernard Cohen. New York: Abaris, 1981.

3. Ismaelis Bullialdi. Astronomia Philolaica. Parisiis: Simeonis Piget, 1645.