ABSTRACT

‘The planets move faster the nearer they are to the Sun, and more slowly the farther they are away from it.’ This fundamental axiom of Copernican cosmology, which Kepler had already adopted in the main in his Mysterium Cosmographicum for the whole planetary universe and had even extended to the motion of individual planets, may be formulated much more accurately, provided always that it is applied only to each individual planet, and not to them collectively. The velocity of a planet in its orbit is inversely proportional to its distance from the body about which it revolves. This, in Kepler's mind, is at first only another statement—yet, how much more reasonable!— of the Ptolemaic theory, which asserts that motion on a deferent is faster the farther the moving point (centre of the epicycle) is from the punctum aequans, and slower when it is nearer; and even that its speed is strictly proportional to its distance from that point. 1 Ptolemy, it is true, made this assertion only in respect of velocity at the apsides. Kepler, however, extended this relationship to the whole path of the orbit; and did so, as historians have not failed to remark, without justifying this extension by any kind of proof; nor was he even aware of the necessity for doing so. It seems to me, that this happened not only in virtue of the ‘continuity principle’, of which he made ample use in his mathematical work, but also, and perhaps especially, because Ptolemy in his view was providing a fact: at the apsides, that is to say under optimum observational conditions, the velocities of planets are proportional to their distance from the central body. This fact agreed so well with his dynamical principles (celestial physics) and Tycho Brahe's observations, that he could not regard it as anything but a general relationship; and if this relationship were not fully confirmed by Ptolemy for other points on the path, in other words, if the agreement between his theory and Ptolemy were only approximate, then the discrepancy was in fact too small to be used as a valid objection against him. There was nothing surprising about this discrepancy; it merely showed that Ptolemy's mathematical technique was not—how could it be?—adequate for the celestial dynamics he was then engaged in developing; and that it was therefore necessary to develop another, based not on abstract concepts, such as the equant, which are fundamentally spurious, but based on reality itself, whose structure must be reproduced. 2 Now, if we abandon the purely kinematic view of Ptolemy, as well as belief in the solid spheres of Copernicus, which Tycho Brahe rendered impossible—these are, in any case, adaptable only to truly uniform motion, and consequently imply the re-introduction of epicycles into the mechanism of planetary motion; and if, therefore, we must seek an explanation of the truly non-uniform motion of celestial bodies, then we must assume, because the velocities of the planets are inversely proportional to their distance from the Sun, that the same relationship holds in respect of the motive forces 3 which drive, or carry, them on their path. So, in Kepler's own words:

‘I have already said that Ptolemy, being led thereto by observations, had bisected the eccentricity of the three superior planets, and that Copernicus copied him. As a result of Tycho Brahe's observations, bisection was shown to be probable also in the case of Mars, as we have described in chapters XIX and XX and as will appear more clearly in chapter XLII. Furthermore, in regard to the Moon, Tycho Brahe proceeded more or less in the same way. Bisection has just been proved in respect of the theory of the Sun (according to Tycho Brahe), or of the Earth (according to Copernicus). There is no objection to the same assumption in the case of Venus and Mercury.… All planets have this bisection. Eight years ago 6 (it is even more, now) in my Mysterium Cosmo-graphicum I postponed [to a later date] a discussion of the cause of the Ptolemaic equant 7 only because it was impossible to decide, from the principles of astronomy as commonly accepted, if the Sun (or the Earth) required a punctum aequans as well as bisection of the eccentricity. This question must now be regarded as settled, particularly as we have confirmation through the evidence of improved astronomy [of the fact] that the Sun, or the Earth, have also a punctum aequans. This being so, I say that the cause of the Ptolemaic equant which I pointed out in my Mysterium Cosmographicum may be taken as true and correct, seeing that it is general and is the same for all the planets. In this part of my work, I shall explain the matter in greater detail.