Kepler, after six years of meticulous study on the orbit of Mars and nine thousand large pages of calculations plotted with his fine writing (1) gets to find out, based on the assumption of circular orbits of the prevailing planets, creating a discrepancy of eight minutes between the positions of Mars and those actually observed that Mars would instead have to deal according to his calculations.
" ... if I thought they could ignore those eight minutes, I would have accommodated as befitted my guess. But it was not allowed to ignore them, those eight minutes pointed the way for a comprehensive reform of astronomy ..." (2)
eight minutes of the episode convinced Kepler that your specific problem - the orbit of Mars - was destined to be insoluble, unless we abandon the tenets of astronomy: the uniform motion, which had already discarded, and now also needed to get rid of circular motion.
" The conclusion is quite simply that the way the world is not a circle - it is curved on both sides and expands in the other two. This curve is called an oval. The orbit is not a circle but an oval shape . "(3)
With the incredible modesty that characterizes certain scientists, Kepler did not feel a hero in having destroyed the dream of the 'harmony of the spheres." Indeed , all you had to say in his defense was:
" I clean the Augean stables of astronomy cycles and spirals, and I left behind me a shitload of manure ." (4)
This
shitload of manure - the motion not uniform circular orbits not - could be explained and justified only by arguments of physics, and geometry (5).
PS At the moment everything what I have available at the handwriting is the description of "fine writing" (as you can imagine).
1-5 Adapted from A. Koestler, The act of creation , Astrolabe Publishing House, pp. 118-119.