On the paradoxical and subsequent course of the Titius-Bode law

Johann Daniel Titius (1729-1796), professor of physics at the former University of Wittenberg (Saxony) translated into German the work Contemplation de la Nature, from the Swiss author Charles Bonnet (1720-1793).

Without saying anything to anybody, Titius inserted two own paragraphs that are at the bottom of page 7 and at the beginning of the 8 in the German edition of 1766. In the preface, Bonnet warns without accurate that Titius has interspersed some own notes, which suggests not only their knowledge but also their conformity. Of course, the new intercropping paragraph is not founded in the original nor in translations of the Bonnet's work in Italian and English.

In the intercalated text we have refer there are to two parts, one after the other. In first part is exposed the succession of planetary distances from the Sun of historical planets, from Mercury to Saturn, rounded to whole numbers as exposed: If we give 100 points to Saturn and 4 to Mercury, to Venus will correspond 4 + 3 = 7 points; to the Earth 4 + 6 = 10; to Mars, 4 + 12 = 16; to the next would be 4 + 24 = 28, but with no planet; and will be 4 + 48 = 52 points and 4 + 96 = 100 points respectively, Jupiter for the and Saturn for the  second.

In the second interleaved part is added as follows: If give to the Earth radius's orbit the value of 10, the other orbits radii are given by the formula Rn = 4 + (3 + 2^n), where n = -∞ for Mercury and 0, 1, 2, 3, 4 and 5 for planets that follow.

These both statements, for all their particular typology and the radii of the orbits, seem to stem from an antique cossist[1]. In fact, many precedents have been finding up to the seventeenth century. Titius was a disciple of the German philosopher Christian Freiherr von Wolf (1679-1754). The second part of the inserted text is also literally founded in a von Wolf's work dated in 1723, and that's why, in twentieth century literature about Titius-Bode law, usually is assigned as authorship the German philosopher; In fact, Titius was a disciple of Wolf. Another reference, older than before, is written by James Gregory in 1702, in his Astronomiae physicae et geometricae elementa, where the succession of planetary distances 4, 7, 10, 16, 52 and 100 becomes a geometric progression of ratio 2. This is the nearest Newtonian formula, which is also contained in Benjamin Martin and Tomas Cerdà himself many years before the German publication of Bonnet's book.

As we have read (3), text interspersed by Titius in Bonnet's book was really transmitted in astronomy's work of Johann Elert Bode (1747-1826). In none of the issues appears Titius, and the authorship of the law is not clearly assigned (Aleitung zur kenntnis des gestirnten Himmels, 1722). Only in a posthumous Bode's memoir can to be founded a reference to Titius with the clear recognition of their priority. But in that moment, everyone knew what was Bode’s Law.

Titius and Bode hoped that the law would lead to the discovery of new planets. But it really was not. Those of the Uranus and Ceres rather contributed to the fame of the Titius-Bode law, but not around Neptune and Pluto's discovery, just because both are excluded. However, it is applied to the satellites and even currently to the extrasolar planets (5).

Titius-Bode law remains a solid and convincing theoretical explanation of their physical meaning, and is not considered as a numerical device. Its history has always been linked as more soup than substance. How can it be compared to the Hipparchus's work in respect to the planetary distances, those of Kepler regarding the orbit of Mars, the discovery of Neptune, the calculation of an event, those of an orbit starting by only three positions, or the explanation about the Mercury's perihelion deviation? However, it is usually more cited.

In the nineteenth century, the Bode or Titius-Bode law: 1) Many authors or do not know, or never cite (9.4). 2) Others use it as if it were a basic law of the celestial mechanics, unrelated to Newton. 3) There also are those who consider it an arithmetic casual approach, and 4) as established as law by Kepler, but with no demonstration.

It is interesting to mention the magnificent book titled The modern telescope from A.T. Arcemis, more than 1,500 pages in two great volumes, published in 1878 (Muntaner & Simon. Barcelona). That book explains to us that Titius’s law has its origin in a French book written by that German author and called Contemplation of nature. It has been needed to reach till XXI century to clarify better this issue, as described in (3).

[1] The cossists were experts in calculations of all kinds and were employed by merchants and businessmen to solve complex accounting problems. Their name derives from the Italian word cosa, meaning “thing”, because they used symbols to represent an unknown quantity, similar to the way mathematicians use x today. All professional problem-solvers of this era invented their own clever methods for performing calculations and would do their utmost to keep these methods secret in order to maintain their reputation as the only person capable of solving a particular problem. 










An explanation of the Titius-Bode’s law that could be previous to its historical origin

The Jesuit Tomàs Cerdà (1715-1791) gave a famous astronomy course in Barcelona in 1760, at the Royal Chair of Mathematics of the College of Sant Jaume de Cordelles (Imperial and Royal Seminary of Nobles of Cordellas). From the original manuscript preserved in the Royal Academy of History in Madrid, Lluís Gasiot remade Tratado de Astronomía from Cerdá, published in 1999, and which is based on Astronomiae physicae from James Gregory (1702) and Philosophia Britannica from Benjamin Martin (1747). In the Cerdàs's Tratado appears the planetary distances obtained from the periodic times applying the Kepler's third law, with an accuracy of 10^-3. Taking as reference the distance from Earth as 10 and rounding to whole, can be established the geometric progression [(Dn x 10) - 4] / [Dn-1 x 10) - 4] = 2, from n=2 to n=8. And using the circular uniform fictitious movement to the Kepler's Anomaly, it may be obtained Rn values ​​corresponding to each planet's radios, which can be obtained the reasons rn = (Rn - R1) / (Rn-1 - R1) resulting 1.82; 1.84; 1.86; 1.88 and 1.90, which rn = 2 - 0.02 (12 - n) that is the ratio between Keplerian succession and Titus-Bode Law, which would be a casual numerical coincidence. The reason is close to 2, but really increases harmonically from 1.82.

The planet's average speed from n=1 to n=8 decreases when moving away the Sun and differs from uniform descent in n=2 to recover from n=7 (orbital resonance).

What announced the Titius-Bode’s law was previously established by Kepler

Among other authors, Comas Solà defined the Titius-Bode law half of the twentieth century as an arithmetic formula that establishes approximately a geometric progression of ratio 2 with the distances of the planets from the Sun, as it had already been established by Kepler previously (4.5).

Like other astronomers of his time, Kepler already had the distances of the planets from the Sun relative to the Earth determined by trigonometric methods (4.1). He also knew the periodic times directly seeing that ratio Pn / Pn-1~ 2, missing a planet between Mars and Jupiter (n = 5). Among n = 3 and n = 7 the distances from the Sun were an exponential function of the sequence of n. Naturally, by the third law, DK = (Pn /P3)^2/3 also knew that (10 x DK)n – 4 / (10 x DK)n-1 – 4 = 2. If he calculates the abnormality he had the uniform circular movement to the real elliptical and could calculate the corresponding radii Rn. This relationship is obtained

Rn – R1 / Rn-1 – R1 = 2 – 0,02 (12-n)

and perhaps it is the most important that exists between Titius-Bode’s law and what Kepler just established previously about it.


Titius-Bode’s law can only be applied to the historical planets of the Copernican solar system from n = 2, and it is not valid for asteroids nor comets

In Figure 17 we have an image, page 10 to the original facsimile of Copernicus’s manuscript (1539) of his famous De Revolutionibus Orbium coelestis. In Figure 16 we find the representation of N. Winston included on page 5 of the Tratado de Astronomía from T. Cerdà T. (1760), comprising planets and comets with different periodic times and distances from the sun.

In Figures 14 and 15 we have the image of the manuscript named Tratado de arismética y geometría práctica from the author Juan de Área y Quiroga (1718, Figure 18), representing the Ptolemaic system (s. II AD.) and Tychonic system (s. XVI AD.) as answer to the question 36 and which would not be applied to the Titius-Bode law.

As we have referred (6.1), each planetary orbit is defined by seven elements among which are the distance to the sun. With the mechanical of Newton it is possible to calculated all the elements from three apparent positions of the planet, with its ascension, straight and decline. We know that this knowledge was historically achieved by successive approximations of many apparent positions, followed by the corresponding adjustments. This is how Kepler and other observers already had much correct orbits of the historical planets.

Comets and asteroids do not follow the Titius-Bode law. Disregarding her, in 1850 Francisco Verdejo Paez, professor of geography at the University of Madrid, in his Geografía Histórica (Imprenta de Repullés, Madrid), makes a unique sequence of distances from the sun, including asteroids known at this time: Mercury, 0.4; Venus, 0.7; Earth, 1.0; Mars, 1.6; Flora, 1.9; Vesta, 2.4; Iris, 2.4; Metis, 2.4; Hebe, 2.4; Astrea, 2.4; Juno, 2.7; Ceres, 2.8; Shovels, 2.9; Higia, 3.0; Jupiter, 5.3; Saturn, 9.7; Uranus, 19.4; and Neptune, 40.5. The inclinations of the orbit from Flora to be Higia: Flora, 5 ° 53 '; Vesta 7 ° 8 '; Iris 5 ° 28 '; Metis, 5 ° 35 '; Hebe, 14 ° 48 '; Astrea 5 ° 19 '; Juno, 13 ° 4 '; Ceres 10 ° 37 '; Palas, 34 ° 38 'and Higia, 3 ° 48'. The author J. Regueiro Argüelles, cited above, in his Astronomia física written at the same year, leaves the distance corresponding to n = 5 with no sun planet, as the latest top authors, and only seven terms for the Bode law.


Interplanetary distances. Geometric mean ratio rn=1,86 from n=3 to n=7 of the Sun distances succession. The most paradoxical success of the Titius-Bode’s law. Influence of the periscope’s introduction

The projected position of the planets in the sky from Earth changes as a result of the simultaneous movement of the Earth and the planet. For example, Earth gives twelve laps around the Sun an Jupiter gives only one, and that’s why we see the planet running half year in one direction and another half on the other, the first faster than the second. There is a circle that is set with the planet revolving twelve times, giving the full revolution to the Sun. This is called epicycles. Those of Saturn is the slowest and Mars the fastest. Those from Venus and Mercury are more complicated. In addition, each planet comes to meet nearer of farther from Earth. For example, at certain points, Mars is four times farther away from us than in others. To explain these variations, it was necessary to modify the circle, from which emerged the idea of ​​a displaced centre of rotation, with the Earth farthest or nearest. This is the eccentric. With Kepler's laws all this was changed. With its calculation of the anomaly it is obtained a circular uniform movement equivalent (4.5), between n = 1 and n = 7, with an average ratio (Rn - R1) / (Rn-1 - R1) = 2 - 0,02 (12, n) = 1.82; 1.84; 1.86; 1,88; 1,90 / 5 = 1.86 instead of 2 as an average value.

For the author of this essay, it is implausible to believe that Copernicus, Galileo, Kepler and Newton were able to interject an anonymous text in the translation of a book as apparently did Titius, and also that they could appropriate this text inserted in an own book, as did Bode. Even today it seems paradoxical the continued success of this action, after three centuries, of what may simply be a relic cosist from early eighteenth century. There are remote causes of these phenomena. In this case we could say that, without the introduction of the telescope by Galileo, the Titius-Bode law would not have existed, but would still implicit in Kepler's work. It has also been said that Copernicus could pull the Earth from the centre of the world thanks to the discovery of America. However, the catalogue of nebulae from Bode and its new constellations could have maligned anyone. But really, what it is happened is that no one remembers about that.

I do not think that rationality and, in particular, scientific knowledge has changed the average of the previous human nature, as it did, for example, passing from the Stone Age to the Age of metals. Homo sapiens stage starts with the writing, which is a relatively recent fact, only about 5.000 to 6.000 years. In the remote darkness of time is Homo habilis, about two million years ago. Iron Age changed as much as we have changed with science. Nothing to do with the small and slow changes that may come after knowing that the earth is a planet revolving around the sun, but eventually no one knows what can happen in the future.

"After having spent more than half a century, the author wrote this essay to the memory of his former teachers."