A Note on the Occasion of the 300th Anniversary of Kepler’s Discovery of the Third Law of Planetary Motions

Priv.-doc. A. I. Bachinsky.

§ 1. Kepler’s first two laws (the ellipticity of planetary orbits and the direct proportionality between the area described by the planet’s radius-vector and the time) were published by him in 1609, in a book entitled: Astronomia nova αἰτιολογητος1, seu Physica Coelestis tradita commentariis de motibus stellae Martis. Ex observationibus G. V. Tychonis Brahe. Jussu et sumtibus Rudolphi II, Romanorum Imperatoris &c. &c. Plurium annorum pertinaci studio elaborata Pragae a Sae Cae Mtis Mathematico Joanne Keplero. Cum ejusdem Cae Mtis privilegio speciali. Anno aerae Dionysianae MDCIX. At first Kepler discovered the so-called “law of areas”; at that time Kepler still held the opinion that the planets move in circles, but that the sun is not at the center of these circles. A geometrical derivation of the law of areas from the propositions of Ptolemaic astronomy is contained in chapter 32 of the Astronomia nova. In the “argumentum” of this chapter this law is formulated in approximately the following words: the intervals of time required for the planet to traverse equal arcs of its orbit are proportional to the planet’s distance from the sun*2. In his figurative language Kepler adds: “prick up your ears, physicists: for here an attempt is being undertaken to invade your domain.” This phrase shows how clearly Kepler understood that the law he had discovered provides a thread for a physical (“causal,” αἰτιολογητος) explanation of planetary motions. Also of interest is the title of chapter 32, devoted to this law: “The force that moves the planets in a circle diminishes with distance from the source.” Kepler believed that any motion not maintained by an external force must die away, and that the revolution of the planets around the sun does not cease only because the sun moves the planets around itself; in order to explain such action on the part of the sun he assumed that the sun is an enormous magnet; the fact that the sun does not attract the planets to itself, but makes them describe orbits, he compared with the way in which a magnet under certain conditions merely directs an iron strip parallel to its “watercourse,” but does not attract it to itself3. Finding that the motion of the pla—

moves along its orbit unevenly, and that it slows down as the planet recedes from the sun, Kepler naturally concluded that the sun’s action upon the planet weakens with distance (but, of course, such a conclusion, by no means was erroneous, though it was built on incorrect premises). Kepler gives the exact formulation (not differing from the modern one) of the law of areas in chapter 40 of his work; and it is curious that he arrives at it by a deductive path from the earlier, incorrect formulation. It goes without saying that from a false premise one can arrive at a true conclusion only in the event that the second premise is also erroneous. Such an erroneous proposition in Kepler is the following: the area of a sector may be regarded as the sum of all the radius-vectors contained within it1.

§ 2. Usually a scholar presents for publication the results he has obtained in that connection of thoughts which in fact led him to these results. He keeps silent about his mistakes, about the false paths onto which he happened to stray, like a traveler in an unknown locality, about fruitless searches, about his enthusiasms for a suddenly flashed new idea, about the enchantments and disappointments he experienced; often even the true primary source of a discovery is artfully masked and replaced by an ad hoc argumentation devised later, better satisfying the requirements of logic than that unexpected, sometimes weakly motivated intuition which often leads to important discoveries.

Not so with Kepler. Kepler, one may say, admits us into the laboratory of his mind, concealing not a single one of its nooks. He sincerely initiates us into all the details of his successes and his failures, is not ashamed to share his emotions with the reader; now he triumphs on the occasion of conquests made, now he good-naturedly laughs at himself over the errors to which he had paid tribute. That is why the reading of his Astronomia nova is so instructive; and that is why there is perhaps no other name in the history of the physico-mathematical sciences that would arouse in us such lively sympathy. Kepler compels us to relive with him the whole genesis of his ideas.

Kepler’s persistent efforts to find such a circumference, motion along which would exactly reproduce the observed positions of the planet Mars, were at times—as it seemed to him—crowned with success. But, in the end, the discrepancies between the theoretical and the actual positions of the planet proved too great to be explained by errors of observation. Then he decides to reject the premise which had behind it the authority of all earlier astronomers and philosophers: the premise of the circular shape of planetary paths2. He tries various ovals; in doing so he needs to measure the areas of these ovals, and he seeks (p. 47) such a θεὸς ἀπὸ μηχανῆς (deus ex machina) who would teach him

to carry out such a quadrature. The matter would have been simple if the orbit had been a perfect ellipse; and he begins to test the elliptical hypothesis. But the results of the tests are unfavorable. Then Kepler, abandoning the idea of an ellipse, again begins to vary the form of the orbit and its position relative to the sun; stubbornly reflecting on one and the same thing, he almost reaches madness (see the end of ch. 58); but, finally, the idea that flashed suddenly in his head reveals to him that, by admitting an elliptical orbit, all the apparent discrepancies can be reconciled; from this moment the elliptical theory of the planets is established unshakably.

§ 3. From 1595 on Kepler diligently sought those simple mathematical relations which, according to his preconceived opinion, ought, in the spirit of Pythagorean doctrine, to bind the number and magnitude of the planetary paths and the motions of the planets along these paths into a single harmonious whole. In his youthful work Prodromus dissertationum cosmographicarum, continens mysterium kosmographicum... (1596) he tries to connect the dimensions of the planetary orbits with the dimensions of spheres circumscribed and inscribed in a known way with respect to the five regular polyhedra. He sets forth the same spatial dependence in his later work Harmonices mundi libri V, which appeared in 1619. Here he also devotes much space to the search for musical ratios among the various characteristics of planetary motions. All this is extraneous speculation. But amid the multitude of calculations having neither a theoretical foundation nor the necessary empirical justification, there shines, in chapter 3 of Book V of the Harmonices¹), like a bright star among clouds, the third law of planetary motions. Kepler prefaces the exposition of this law with an account of the first two laws, formulating them both more precisely and with stronger emphasis than had been done in the Astronomia nova. I take the liberty of giving here a translation of the corresponding passage from the Harmonices:

“... In order to pass on to the motions, among which harmony is established, I again remind the reader that in the notes on Mars I proved, on the basis of Tycho’s extremely precise observations, that equal daily²) arcs on one and the same orbit are traversed not with equal speed, but that these different durations [the planet’s stay] on equal parts of the orbit preserve proportionality to the distances of these parts from the sun, the source of motion; and that, in turn, if equal intervals of time are assumed (for example, one natural day in either case), the corresponding true daily arcs of one orbit are inversely proportional to the distances from the sun³). At the same time I proved that the orbit of a planet is an ellipse, and the sun, the source of motion, is located at one of the foci of this ellipse; and it happens that the planet, having completed a quarter of the whole circuit from its aphelion,

¹) This chapter bears the following title: “The chief points of the astronomical doctrine necessary for understanding the celestial harmonies.” The whole book is entitled: “On the most perfect harmony of the celestial motions and on the origin, from it, of the excentricities, semidiameters, and periods of revolution.”

²) The word “daily” here, of course, should not be understood literally; Kepler means: small arcs corresponding approximately to an interval of time of 1 day.

³) This formulation, like the preceding one, is not entirely precise. Further on Kepler himself stipulates the conditions under which it is applicable.

(Notes by A. Bachinskaya).

is separated from the sun by such a distance as is precisely the arithmetic mean of the greatest distance at aphelion and the least at perihelion. And from these two axioms it follows that the mean daily motion of the planet along its orbit is just as great as the true daily arc of that orbit at those moments when the planet is at the end of a quarter of the orbit, counting from aphelion (although this true quarter appears smaller than an angular quarter). Further it follows that two true daily arcs of the orbit at any equal distances, one from aphelion and the other from perihelion, are together equal to two mean daily [arcs]1... This is what must first be learned about the true daily arcs of the orbit and about the true motions, so that from this we may then understand the apparent motions, assuming the eye to be in the sun.

But as for the apparent arcs2 from the point of view of the sun, it is already known from ancient astronomy that of the true motions, even those equal to one another, the one farther from the center of the universe (as being at aphelion) appears smaller to an observer at the said center; while the one nearer, as being at perigee, appears larger. And since, in addition, the true daily arcs near the latter prove to be larger, owing to the more rapid motion, while at the remote aphelion, owing to the slowness of the motion, they are smaller, from this I have proved in the book on Mars that the ratio of the apparent daily arcs of one orbit is, with sufficient accuracy, equal to the square of the inverse ratio of their distances from the sun3. Thus, if a planet on one of its days, being at aphelion, is distant from the sun by 10 parts (in some measure), and on the opposite day, being at perigee, by 9 such parts, then there is no doubt that, as seen from the sun, its apparent motion at aphelion will stand to its apparent [motion] at perigee as 81 to 100.

But this is true with the following precautions: first, that the arcs of the orbit should not be large, lest they be associated with different distances that differ appreciably [from one another], that is, lest they involve a noticeable difference in the distances of their ends from the apsides; then, that the eccentricity should not be too large...

Up to now we have been speaking of different intervals of time and [traversed] arcs for one and the same planet. Now, however, we must turn to the motions of two planets compared with each other... Thus here I must complete and introduce into this place a certain part of my “Cosmographic Mystery,” left unresolved 22 years ago, since at that time the matter was not yet clear to me. And so, after—

(Notes by A. Bachinsky.)

by continuous labor over a very long time, from Brahe’s observations were found the highest intervals of the orbits, finally, finally, the true proportion between the periods and the sizes of the orbits

...though late, she noticed the idle one,
Noticed him nonetheless and came, after a long interval of time ¹);

and if an indication of the time is desirable—she was born in my mind on March 8 ²) of this one thousand six hundred and eighteenth year, but was unsuccessfully calculated and therefore rejected as false; but when on May 15 I returned to her, taking her up with renewed enthusiasm, she at last overcame the blindness of my mind; this was so great a reward for my seventeen-year labor over Brahe’s observations, and for the meditation directed in accordance with them, that at first I was ready to think that I was dreaming and that I was assuming the sought-for among the given. But it is in the highest degree certain and exact that the ratio between the periods of revolution of any two planets is exactly equal to the sesquialterate power of the ratio of their mean distances, that is, of the [radii of the] orbits; however, I draw attention to the fact that the arithmetical mean of the two diameters of an elliptical orbit is somewhat less than the longer diameter. Thus, if someone from the period—let us say—of the Earth, which is equal to one year, and from the period of Saturn (thirty years) takes the cube roots and, squaring these roots, forms a ratio, then he will have in the resulting numbers the truest ratio of the mean distances of the Earth and Saturn from the Sun.

For the cube root of 1 is 1, its square is 1; and the cube root of 30 [and something] is more than 3, its square [and something] is more than 9. And Saturn, at its mean distance from the Sun, is somewhat farther than nine times the mean distance of the Earth from the Sun.”

The exposition of the third law and of its discovery, permeated with special rapture, shows that Kepler assigned to this law an outstanding role among his achievements. It must be noted, however, that here the role was played not so much by this law in itself as by the harmonic relations in the solar system that Kepler attached to this law ³).

¹) Kepler quotes the Latin verses:

sera quidem respexit inertem,
Respexit tamen et longo post tempore venit.

At the same time he, with a certain humor, depicts the matter as though the law he had discovered were an independently living idea which found him, Kepler—and not the reverse.

²) This date, apparently, is given according to the Gregorian calendar (which in the sixteenth–seventeenth centuries differed from the Julian by 10 days).

(Note by A. Bachinsky.)

³) To assess Kepler’s own notions of these harmonic relations, the preface of Kepler to Book V of the Harmonices is characteristic. Here is this preface:

“What I divined twenty-two years ago, as soon as I discovered five regular solids among the celestial circles; of which I was firmly convinced earlier, as soon as I saw Ptolemy’s Harmonics; what I promised to friends, having announced the title of this fifth book before, before I myself obtained the final result; what I promised and sought out, what I had firmly in mind for sixteen years before this; for the sake of which I devoted the best part of my life to astronomical investigations, turned to Tycho Brahe, chose Prague as my place of residence—that, at last, by the grace of God the all-good, the Most High (who inspired my mind, aroused a mighty desire, prolonged my life and powers of soul, and provided other means through the generosity of two emperors and nobles of Upper Austria), having advanced from the dawn of exact astronomical activity, I, finally, I say, have appeared this before ...

On the character of these relations one may judge from the titles of the separate chapters composing Book V of the Harmonices; here are these titles:

“I. On the five regular polyhedra.

II. On the kinship with them of harmonic relations.

III. The principal point of the astronomical doctrine, necessary for understanding the celestial harmonies.

IV. In what circumstances, relating to planetary motions, simple harmony is expressed; and that in the heavens are found all those which are sung in song.

V. The steps of the musical scale and the kinds of harmonies—major and minor—are expressed by certain motions.

VI. The individual musical modes, or tones, are expressed by individual planets.

VII. There may exist counterpoints, or universal harmonies, of all the planets, and these in various kinds, but one flowing from another.

VIII. In the planets are expressed the natures of the four voices—descant, alto, tenor, and bass.

IX. Proof that, in order to obtain this harmonic arrangement, the eccentricities of the planets must have been established in such a way that each of them has its own, and no other.

X. Epilogue on the Sun, on the basis of very weighty considerations.”

§ 4. It is known that if to the three Keplerian laws of planetary motions one adds three general dynamical principles: the principle of inertia, the principle of the independent action of forces, the principle of action and reaction,—then from this, by a not particularly complicated induction, there is obtained the Newtonian law and, in final form (beyond what Galileo had already given before Kepler), that whole nature of harmony, so variegated and multifarious, and all those laws of the planetary system and their many consequences which nature has disclosed to us, and in which, as I have already remarked, this side constitutes for us a special grandeur, but in a wholly new, at once most remarkable and most modern sense. At that vague time when I was restrained in doubt by the extremely difficult reproduction of the phenomena, it occurred to me, to my particular joy, to make a passionate pronouncement and to stir up a great dispute in Palm’s Treatise, including the Harmonices. Ptolemy (with the invention of musical ratios) was assigned the foremost place, the middle place to him who applies these ratios to all manner of theory, the place after Ptolemy being occupied by Bullialdus; on which I, from above, with expectation and the greatest amazement, shall write that almost all his three laws were posited fifteen hundred years earlier. To the same investigator of the motions of the heavenly bodies, Copernicus, much else can be justified from the law of uniformity, and Ptolemy may discuss the unequal or wandering motions of the others, just as we show in the sequel, that the same opportunity was open to Ptolemy with Hipparchus, rather than the glorious Pythagorean doctrine should bring benefit to philosophy; but even in me, by the imperfections of ancient astronomy, to this very thing, to an exact agreement in the intervals of two persons, separated by an interval of fifteen centuries, in the highest degree it confirmed and established the presupposition (which I at the same time most warmly and in many places expressed in my Harmonices) of the exact and true agreement in the length of the intervening intervals in the extent of their concepts; so that a dispute arose among scholars about how, here in the minds of two men who had entirely grasped the laws of nature, one and the same conception of the universe was formed, although neither of them was constrained by the other when entering upon this path. After you have spent a month naming that first term, and, after two months, set out on the path, and after a few more days find the very exact relation of the Pythagorean ratios, nothing will hold me back any longer: I want to give free rein to sacred frenzy, I want to laugh at mortals, openly confessing that I have stolen the golden vessels of the Egyptians in order to build from them a tabernacle to my God, far from the confines of Egypt. If you forgive me, I shall rejoice; if you are angry, I shall endure it; here I cast my die and write a book which will be read by contemporaries or posterity—either makes no difference; let it await its reader for a hundred years, if God has waited six thousand years for an observer.”

the law of gravitation between the Sun and the planets. In the history of physics the question has often been raised: how close was Kepler to carrying out such a deduction? It is clear that, in order to clarify this question, it is necessary first to determine in what form and to what extent the true dynamical principles were clear to Kepler. The material for the corresponding determination is found in Astronomia nova, as well as in Kepler’s third major astronomical work, bearing the title Epitome Astronomiae Copernicanae and published in parts during the years 1618–16211. In the introduction to Astronomia nova Kepler briefly and clearly sets forth his fundamental dynamical ideas in the following words:

“True doctrine concerning gravity is based on the following axioms2:

1. Every corporeal substance, insofar as it is corporeal, is by nature adapted to rest in any place in which it is placed alone, outside the sphere of action of a kindred body.

2. Gravity is a mutual corporeal disposition toward union between kindred bodies (to which order of things the magnetic faculty also belongs), so that the earth attracts a stone much more than the stone strives toward the earth.

3. Heavy bodies (especially if we place the earth at the center of the universe) do not strive toward the center of the universe as toward the center of the universe, but as toward the center of a round kindred body, that is, the earth. Thus, wherever the earth might be placed or wherever it might be carried by its animate faculty, heavy bodies would always strive toward it.

4. If the earth were not round, heavy bodies would not strive from everywhere directly toward the middle of the earth, but would strive from different sides toward different points.

5. If in some place of the universe two stones were placed close to one another, outside the sphere of action of a third kindred body, then these stones would come together, like two magnetic bodies, at some intermediate place, each approaching the other by such a distance as is large in comparison with the massiveness of the other.

6. If the moon and the earth were not held, each in its orbit, by an animal or some other equivalent force, then the earth would rise toward the moon through a fifty-fourth part of the distance [between us], while the moon would descend toward the earth by approximately \(53/54\) of the distance, and there they would be joined (assuming that the substance of both has one and the same density).

7. If the earth ceased to attract to itself its waters, then all the waters of the seas would rise and be poured out upon the body of the moon”3...

Let us proceed to an examination of the content of the individual axioms.

1. The first axiom is nothing other than the law of inertia, expressed, however, only, so to speak, halfway. Kepler is entirely unaware that a body “by nature” can not only remain at rest, but also move uniformly in a straight line. (Therefore those authors are mistaken4 who attribute—

Kepler is credited with the discovery of the law of inertia in the form just given. In contrast to this, Kepler believes that “matter, as such, has no capacity whatever to transfer its body from place to place”1).

In this half-and-half form Kepler advances the law of inertia very often2).

The significance of this thesis in Kepler is chiefly polemical—against the Aristotelian school. According to Aristotle, every body naturally strives to occupy its own place; such a place for “heavy” bodies is the center of the universe, and for “light” bodies—its periphery. Motions arising as a result of this natural striving were called natural; every other motion was classed as violent. Every violent motion (for example, the motion of a thrown body) must necessarily die out; on the contrary, natural motions (such as the motion of a freely falling body) are accelerated. It is interesting that in Aristotle, in one place (Physics, book IV, ch. 11), there occurs the notion of such a motion as we call “motion by inertia,” but Stagirite treats this notion as an evident absurdity. The question concerns the impossibility of a void; the argument consists in the following: if a void existed, it would be completely incomprehensible why the velocity of a body moving in the void should change. The body would have either to remain forever at rest, or to move forever, until some obstacle stopped it. And since this does not occur in nature, therefore, it follows, the void does not exist.

Kepler struggles against the notion of a “natural” striving of bodies to move toward the center of the universe (or toward its periphery); that is why he must first of all establish the axiom of the natural inclination of bodies to rest (the very term inertia to denote this inclination was apparently introduced by him).

2–7. The following axioms of Kepler contain an entirely sound doctrine of gravitation3). Especially remarkable is the proposition about the inverse proportionality between the masses and the motions of two mutually attracting bodies (one may say that in this proposition Newton’s law of action and reaction is implicitly contained). Kepler illustrates this proposition with the example of two stones, with the example of the earth and the moon (whereby he takes the ratio of the volumes of the earth and the moon to be 53 : 1 instead of the correct 50 : 1). The phrase in the 2nd axiom might cause some perplexity: “the earth draws the stone much more than the stone strives toward the earth”; however, here the word more refers, of course, not to the magnitude of the action, but to the result of the action of the force—to the motion, exactly as was the case in the examples just mentioned.

We see that, of the dynamical principles, Kepler possessed the first half of the principle of inertia and the principle of action and reaction. The second half of the principle of inertia was completely alien to him; as for the principle of the independent action of forces, Kepler had rather vague notions of it4). It is clear that under such conditions there could be no question even of a correct dynamical inter—

of the law of areas; not to mention the investigation of the two remaining laws of planetary motions, which is more difficult.1 Nevertheless, Kepler managed by intuition to come very close even to the quantitative law of gravitation. In trying, in the Epitome, to explain the mechanical action of the Sun upon the planets, he uses, as models and auxiliary means, now the law of the lever, now magnetic interactions, and finally compares the force issuing from the Sun and acting upon the planets with the rays of light and heat likewise issuing from it; whence he comes to the question: if illumination decreases proportionally to the square of the distance, will not the moving force also be inversely proportional to the square of the distance? However, he answers this question in the negative.2

One may express, as a probable conjecture, that if Kepler had read Galileo’s Discorsi e dimostrazioni matematiche intorno à due nuove scienze attenenti alla Mecanica e ai movimenti locali (where those very dynamical principles are developed with complete clarity, the knowledge of which Kepler lacked), then, with the aid of his genius and his investigative persistence, he would have been able to arrive at a quantitative formulation of the law of gravitation. But death overtook Kepler in 1630, while the Discorsi appeared in print in 1638.

A. Vasiliev.

1. For this reason the authors of the commentary to the French translation of Newton’s Principia are not entirely right when, with a tinge of reproach, they say of Kepler: “Il n’a tiré de ce principe (principle of gravitation) ce qu’on auroit dû croire qu’un aussi grand homme que lui en auroit tiré.” [Principes math. de la Phil. Nat., II, Exposition abrégée du Système du Monde, p. 5 (1759)]. ↩↩↩↩↩↩

2. Opera, t. 6, p. 349. ↩↩↩↩↩↩

3. In the presence of such precedents, there is hardly any need to invoke the legendary apple to explain why Newton began to think about gravitation and its causes. ↩↩↩↩

4. See Opera, vol. 6, pp. 181, 345–346. ↩↩