I am in the process of preparing equations for new three dimensional graphics by Dr. Horst Eckardt in view of the immediate interest in the discovery of three dimensional orbits. Since Dr. John Maddocks mentioned Leibniz, the Leibniz equation of 1689 is:
m r double dot = L squared / (m r cubed) – k / r squared
in which the centrifugal force appeared for the first time (the first term on the right hand side). Huygens inferred the word “centrifugal”, and Newton used “centripetal” but neither Huygens nor Newton derived the mathematical form of the centrifugal force. The second term on the right hand side is the inverse square law. My ancestral cousin John Aubrey wrote in “Brief Lives” that Robert Hooke inferred the inverse square law, well before the younger Isaac Newton. However Hooke did not infer calculus so could not prove that the inverse square law leads to an ellipse. Hooke and Newton disagreed severely about precedent, and Hooke advised Aubrey when the latter wrote “Brief Lives”. Newton and Lebniz disagreed about precedent in the context of calculus. It is generally considered that all three contributed importantly to the subject. The first to derive the non Newtonian accelerations correctly was Coriolis, in 1835. In UFT271 on www.aias.us they are derived for the first time in spherical polar coordinates, resulting in a rich panoply of new information about orbits, and in the acclaimed graphics of co author Horst Eckardt from my equations. Horst Eckardt also checked these equations by computer, as he had done in over two hundred UFT papers. Leibniz derived the correct mathematical format of the centrifugal force by using intuition via a theory that was later replaced by the rotating coordinate method, used in modern format in UFT271. I have read a few books on scientists, but until I got in to the subject matter to sufficient depth, words left me with little understanding. Modern graphics and animations however can turn the most complicated set of equations into something that is immediately understandable. I learnt that as a first year graduate when I was first got access to a computer. I read a book on Newton in Wolfson College Oxford having just won a prestigious Junior Research Fellowship there. One does not really get to grips with orbit theory with Newtonian dynamics, because the spin connection is missing. The axes are not rotating, so in Newtonian dynamics there is no centrifugal force. Without a centrifugal force there is no orbit. So the often repeated claim that Newton derived the ellipse from the inverse square law is obscure. I also read some books as a graduate about general relativity, but they did not really reveal anything until I started to use ECE theory. The best biographical work that I came across is Koestler’s “Sleep Walkers”, about Copernicus, Brahe, Galileo, Kepler and Newton. Koestler uses the novelist’s technique combined with accurate historical scholarship. However, without the equations, that book too falls short of complete clarity. In those days there were few equations, and on reading Principia in Latin it is immediately apparent that Newton did not use any recognizably modern equations. Neither did Kepler. So I advise readers to follow UFT271 and that will give all the information needed. The excellent graphics by Horst turn the very intricate equations into something that is understandable. A lot of dogma creeps in to physics because the mathematics are taught in none too clear a way. It is only after grinding through the hard work for many years that one achieves an understanding that can be built upon so that new discoveries can be made. Some of these books are good reading, but go so far in words and no further. For example Koestler uses a lot of Kepler’s diary in which he gives every day details such as having trouble sitting down because of boils and all that, and of Brahe having his nose sliced off in a duel. That is amusing and good reading. Both Tycho Brahe and Johannes Kepler were appointed Imperial Mathematicus in Prague. However Kepler was often not paid. Things have not changed very much. Galileo got into trouble with authority, but in fact the Jesuits accepted his theory, he just quarrelled with authority, an entirely healthy thing to do, but in those days dangerous. Copernicus was infinitely careful. Tycho Brahe was a nobleman from Denmark, but Kepler came from Schwabian country people (peasants) and spoke with heavy Schwabian accent. His own University of Tuebingen never appointed him and he had a great deal of trouble getting Brahe to release his data on Mars.