The Cylinder. Helmut Müller-Sievers

status of geometric relations and of the motions they embodied was discussed with renewed enthusiasm when new editions and commentaries on Plato’s Timaios appeared in the fifteenth and sixteenth centuries. One of the key moments in this interpretation occurs in Marsilio Ficino’s commentary on the Timaios from 1496. Commenting on the famous section on the origin of the world-soul (Tim. 36bc), Ficino claims that the natural motion of the soul is translational (animae motum naturaliter esse rectum) and that it is the task of intelligence (which itself is a gift of God) to bend it into rotational (in gyrum) motion.²⁵ The mysterious relation between the power of straight lines and angles and the nimbus of the sphere finds, as mentioned, a striking expression in Kepler’s Mysterium cosmographicum, where the Platonic solids (composed of regular rectilinear modules) are encapsulated in ever-larger spheres to demonstrate the distance between and the orbital motion of the planets. Copernicus earlier had given a succinct summary of the metaphysics of rotation and sphericity when he stated that the sphere is the perfect form because it is without “joint” and that everything that limits itself—a drop of water, for example, but also the sun and the planets—does so in the form of a sphere The motion appropriate to this perfect form is, of course, rotation.²⁶

      All the trust put into the power and rationality of the straight line provided the ground for the assertion, first tentatively by Galileo and Descartes, then exhaustively by Newton, that motion along a straight line is the natural motion of any body in the universe. A corollary of this assertion is that space must be conceived as empty, homogenous, and infinite, since otherwise this motion would come to an inexplicable end. Alexandre Koyré has eloquently described the stages in this transition from the spherical cosmos to the infinite universe.²⁷ But the full acceptance of the translational motion paradigm came with some hesitations, and the objections all had to do with the nature of rotation. Although he established the idea of uncaused, inertial motion, Galileo for one could not convince himself that the orbits of the stars were just the product of two conflicting linear motions. His adherence to the Platonic idea of rotational and spherical perfection led him to reject the idea of a universe in which inertial motion could be conceived only as translational.²⁸ For Descartes, cosmic vortices carried planets around their axis, taking everything around with them into rotation.

      Newton’s “great synthesis,” as we have seen in the discussion of Kleist’s text, was based on a previous analysis, namely the drastic separation of kinetic phenomena from the aesthetic and theological considerations that had dominated scholastic science and theology and that still left traces on early modern physics. Some motions are not “better” or “more beautiful” than others, Newton declared; they are simply the result of the measurable impact of forces on mass.²⁹ With the concept of mass Newton could abstract from any shape or position and extend calculations beyond the reach of the observable. One might not know what distant stars look like, but one could be sure that they were composed of quantifiable mass because its effect—gravitational pull—was measurable in their orbits. This abstraction, together with the great distances involved in celestial mechanics, made it possible to treat any body as a nonextended point mass: for the purpose of calculation—say, to calculate the gravitational force of the moon—it sufficed to conceive of its mass as being compressed in a point at the center of the physical globe. Newton, an atomist, believed in the irreducible extension and indivisibility of physical bodies, but for the purpose of calculation this philosophical commitment could be disregarded.³⁰ He felt even more justified in reducing celestial bodies to points when he could show—as he did in the debate with the Cartesians over the shape of the earth—that a body of malleable matter rotating in empty space around its central axis would morph into a regular spheroid whose center of mass would coincide with its geometric center. Points, in turn, could become the stuff of geometry—their path could be described in geometric curves with perfect accuracy, and they could become subject to the predictive power of algebraic operations.

      Newton was perfectly aware that there were limits to this mode of explanation; indeed, he was eager to point them out to counter the suspicion that he conceived of a fully mechanized, self-sufficient universe. One such limit was the implication of a void between bodies, and of forces acting across it. For rational mechanics to work, gravity had to act instantaneously and bodies had to be distinguishable from their surroundings; but how could such actio in distans be understood? How could motion change (as it did in Kepler’s elliptical orbits) without any contact? Then there was the related question of whether the distances between the planets, placed as they were at the exact intervals that kept them from collapsing into the center and from flying off into space, could originate through mechanical forces. Newton enthusiastically embraced Bentley’s suggestion that this might serve as a cosmological proof for the existence of God.³¹

      As far as the motion of the planets was concerned, Newton admitted to Bentley that gravity would explain the centripetal factor of the planets’ orbits, “yet the transverse motions by which they revolve in their several orbs required a divine Arm to impress them according to the tangents of their orbs.” Since this did not necessarily include the rotational motion of the planets, Newton added “that the diurnal rotations of the Planets could not be derived from gravity but required a divine power to impress them.”³²

      This cosmological argument had a mechanical counterpart in the fact that, according to Newton’s second law of motion, any change of motion was proportional to the magnitude of a force impacting a body; both the impact and the resulting direction would be in a straight line. How could rotation originate from the impact of just one force?³³

      For reasons like these Immanuel Kant introduced a second “original” force besides gravitation into the fabric of the universe in his daring Allgemeine Naturgeschichte und Theorie des Himmels of 1755: the repulsive force. He blunted the audacity of this addition to Newton’s mechanics by arguing that “these two forces are both equally certain, equally simple, and at the same time equally primal and universal. Both are taken from Newtonian philosophy. The first is now an incontestably established law of nature. The second, which Newtonian science perhaps cannot establish with as much clarity as the first, I here assume only in the sense which no one disputes, that is, in connection with the smallest distributed particles of matter, as, for example, in vapours.”³⁴ To show the primordial interplay of these forces, Kant imagined the world “on the immediate edge of creation,” when the universe was filled with matter at rest for a time “which lasts but an instant.”³⁵ Since atoms were created with different specific weights, the heavier ones attracted the lighter ones and began to form “gobs” (Klumpen.)³⁶ All matter would collapse into one big gob were it not for the repulsive force that inflected the straight path of onrushing matter and sent it into an orbit around the central, that is, heaviest body. Applied to the formation of the solar system, the interaction of these two forces explained why all planets orbited around the sun in one plane—the central mystery for Newton in his exchange with Bentley. They were all remnants of the initial cloud of matter that had first collapsed on, and then been flung from, the heaviest gob in one part of the universe, the sun. The same had happened in countless other corners of the universe.

      This “nebula hypothesis” was a theory with extraordinary explanatory power, justly famous for its range and daring: in one fell swoop it explained the origin of empty space (as the consequence of matter contracting), the spherical form of celestial bodies (as the consequence of the simultaneous rush of particles on a common center and the resulting rotation) and the common plane of all orbits in the solar system, resulting in a fully mechanical cosmogony.³⁷ But subtly it also reversed the question of the origin of rotation. In a later chapter, “Concerning the Origin of Moons and the Axial Rotation of the Planets,” Kant makes the much-needed distinction between orbital motion (“Zirkelbewegung”) and axial rotation (“Achsendrehung”) and explains the origin of the latter as the result of particles impacting the already forming body, off-center and from opposite sides, and thereby keeping it spinning.³⁸ The diameter of the planet serves as a lever on which the particles exert opposite, yet equal translational force. Kant’s hypothesis anticipates here the notion of torque as the product of the length of a lever arm and two opposite perpendicular forces: he argues, for example, that Jupiter rotates faster than smaller planets (like Mars), which can