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The Titius–Bode law (sometimes termed simply Bode's law) is a formulaic prediction of spacing between planets in any given planetary system. The formula suggests that, extending outward, each planet should be approximately twice as far from the Sun as the one before. The hypothesis correctly anticipated the orbits of Ceres (in the asteroid belt) and Uranus, but failed as a predictor of Neptune's orbit. It is named after Johann Daniel Titius and Johann Elert Bode. Later work by Blagg and Richardson significantly revised the original formula, and made predictions that were subsequently validated by new discoveries and observations. It is these re-formulations that offer "the best phenomenological representations of distances with which to investigate the theoretical significance of Titius–Bode type Laws". Original formulation The law relates the semi-major axis   a n   {\displaystyle ~a_{n}~} of each planet outward from the Sun in units such that the Earth's semi-major axis is equal to 10:   a = 4 + x   {\displaystyle ~a=4+x~} where   x = 0 , 3 , 6 , 12 , 24 , 48 , 96 , 192 , 384 , 768 …   {\displaystyle ~x=0,3,6,12,24,48,96,192,384,768\ldots ~} such that, with the exception of the first step, each value is twice the previous value. There is another representation of the formula:   a = 4 + 3 × 2 n   {\displaystyle ~a=4+3\times 2^{n}~} where   n = − ∞ , 0 , 1 , 2 , …   . {\displaystyle ~n=-\infty ,0,1,2,\ldots ~.} The resulting values can be divided by 10 to convert them into astronomical units (AU), resulting in the expression: a = 0.4 + 0.3 × 2 n   . {\displaystyle a=0.4+0.3\times 2^{n}~.} For the far outer planets, beyond Saturn, each planet is predicted to be roughly twice as far from the Sun as the previous object. Whereas the Titius–Bode law predicts Saturn, Uranus, Neptune, and Pluto at about 10, 20, 39, and 77 AU, the actual values are closer to 10, 19, 30, 40 AU. Origin and history The first mention of a series approximating Bode's law is found in a textbook by D. Gregory (1715): "... supposing the distance of the Earth from the Sun to be divided into ten equal Parts, of these the distance of Mercury will be about four, of Venus seven, of Mars fifteen, of Jupiter fifty two, and that of Saturn ninety five." A similar sentence, likely paraphrased from Gregory (1715), appears in a work published by C. Wolff in 1724. In 1764, C. Bonnet wrote: "We know seventeen planets [that is, major planets and their satellites] that enter into the composition of our solar system; but we are not sure that there are no more." In his 1766 translation of Bonnet's work, J.D. Titius added two of his own paragraphs to the statement above. The insertions were placed at the bottom of page 7 and at the top of page 8. The new paragraph is not in Bonnet's original French text, nor in translations of the work into Italian and English. There are two parts to Titius's inserted text. The first part explains the succession of planetary distances from the Sun: Take notice of the distances of the planets from one another, and recognize that almost all are separated from one another in a proportion which matches their bodily magnitudes. Divide the distance from the Sun to Saturn into 100 parts; then Mercury is separated by four such parts from the Sun, Venus by 4+3=7 such parts, the Earth by 4+6=10, Mars by 4+12=16. But notice that from Mars to Jupiter there comes a deviation from this so exact progression. From Mars there follows a space of 4+24=28 such parts, but so far no planet was sighted there. But should the Lord Architect have left that space empty? Not at all. Let us therefore assume that this space without doubt belongs to the still undiscovered satellites of Mars, let us also add that perhaps Jupiter still has around itself some smaller ones which have not been sighted yet by any telescope. Next to this for us still unexplored space there rises Jupiter's sphere of influence at 4+48=52 parts; and that of Saturn at 4+96=100 parts. In 1772, J.E. Bode, then aged twenty-five, published an astronomical compendium, in which he included the following footnote, citing Titius (in later editions): This latter point seems in particular to follow from the astonishing relation which the known six planets observe in their distances from the Sun. Let the distance from the Sun to Saturn be taken as 100, then Mercury is separated by 4 such parts from the Sun. Venus is 4+3=7. The Earth 4+6=10. Mars 4+12=16. Now comes a gap in this so orderly progression. After Mars there follows a space of 4+24=28 parts, in which no planet has yet been seen. Can one believe that the Founder of the universe had left this space empty? Certainly not. From here we come to the distance of Jupiter by 4+48=52 parts, and finally to that of Saturn by 4+96=100 parts. These two statements, for all their peculiar expression, and from the radii used for the orbits, seem to stem from an antique algorithm by a cossist. Many precedents were found that predate the seventeenth century. Titius was a disciple of the German philosopher C.F. von Wolf (1679–1754), and the second part of the text that Titius inserted into Bonnet's work is in a book by von Wolf (1723), suggesting that Titius learned the relation from him. Twentieth-century literature about Titius–Bode law attributes authorship to von Wolf. A prior version was written by D. Gregory (1702), in which the succession of planetary distances 4, 7, 10, 16, 52, and 100 became a geometric progression with ratio 2. This is the nearest Newtonian formula, which was also cited by Benjamin Martin (1747) and Tomàs Cerdà (c. 1760) years before Titius's expanded translation of Bonnet's book into German (1766). Over the next two centuries, subsequent authors continued to present their own modified versions, apparently unaware of prior work. Titius and Bode hoped that the law would lead to the discovery of new planets, and indeed the discovery of Uranus and Ceres – both of whose distances fit well with the law – contributed to the law's fame. Neptune's distance was very discrepant, however, a.... Discover the Christine Bode popular books. Find the top 100 most popular Christine Bode books.