Kepler was not content to propose esoteric schemes for the order of planets in the heavens and to be satisfied if they worked approximately. He believed that any theory should agree with the available data quantitatively. He was aware, as were all the astronomers in Europe, of the great amount of accurate data that had been obtained by Tycho Brahe. Brahe, on the other hand, was aware of Kepler's mathematical ability. In 1600, Kepler became an assistant to Brahe, who had moved to the outskirts of Prague (now in the Czech Republic) and had become Court Mathematician to Rudolph II, Emperor of the Holy Roman Empire (the fanciful name by which the Austro-Hungarian Empire was once known). Kepler and Brahe did not always get along well, but Kepler found Brahe's data a very powerful attraction. Brahe died 18 months later in 1601, and Kepler immediately took custody of all the data. Shortly thereafter, in 1602, Kepler was appointed Brahe's successor as Court Mathematician to Rudolph II.

As mentioned earlier (Section D3), Kepler had been urged by Brahe to try to refine Brahe's compromise theory to fit the data, but he was not successful. He then returned to the heliocentric theory, but did not share Copernicus's objection to the equant (Section C3), so he included that in the calculations. He also made some improvements on some of Copernicus's detailed assumptions, making them

10The Earth even retains some uniqueness in this scheme. Its sphere rests between the two solids having the greatest number of faces, the dodecahedron and the icosahedron.

"Because we now know that there are at least nine planets, this scheme is of no value. The questions may still be asked about why the planets have the spacings they have, and if there is any rule that determines how many planets there are. Another scheme that was once thought valid (but no longer) is described by Bode's law or the Titius-Bode rule. A discussion of this rule is beyond the scope of this book, but a fascinating account of its history and the criteria determining whether such a rule is useful is given on pages 156-160 of the book by Holton and Brush, listed as a reference at the end of this chapter.

Figure 2.10. Elliptical orbits, (a) An ellipse: S and P are the two foci. The sum of the distances to any point on the ellipse from the foci is always the same, (b) Kepler's second law: The planet takes the same amount of time to traverse the elliptical arcs bounding the two shaded areas if the shaded areas are of equal size.

more consistent with a truly heliocentric theory. He concentrated particularly on the calculations for the orbit of Mars, because this seemed to be the most difficult one to adjust to the data. The best agreement he could get was on the average within 10 minutes of arc. Yet he felt that Brahe's data were so good that a satisfactory theory had to agree with the data within 2 minutes of arc. He felt that the key to the entire problem for all the planets lay in that 8-minute discrepancy. Gradually and painstakingly, over a period of two decades, punctuated by a variety of interruptions, including changes of employment, the death of his first wife, and a court battle to defend his mother against charges of witchcraft, he worked out a new theory.

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