and its dimensions are connected to an increasingly linear universe.17
Leon Battista Alberti set this preference for horizontal lines in stone. In his foundational treatise De re aedificatoria he challenged the unfavorable etymology that derived the name of the builder’s profession from the curve (arcus) of the roof (tectum).18 He asserted that rather than celebrating transcendence, as cathedrals do in their height and vertical intricacy, churches as well as representative palazzi and private homes should exhibit strong horizontal lines that converge on the altar, or on doors and windows.19 These lines were understood as guidelines for visual rays on which the objects of vision traveled to and from the eyes. This inherent belief in the coincidence of geometric lines and natural motion found its most confident expression in Galileo’s assertion that the book of nature and its motions was written in the language of geometry.20 The relation of priority that Alberti established between the regularity of geometric proportions and lines and their embodiment in the motion of extended bodies would hold for many centuries, and in many fields. The house of memory, for example—the aid by means of which an orator would memorize the parts of his speech and their sequence—underwent an Albertian renovation: whereas ancient and medieval memory houses had regarded the difference between the rooms and the floors as an aid to memory, in early modern memory houses rooms were differentiated solely by their connection to other rooms.21
Also the active employment of the intellect was conceived as moving along straight lines with regular bifurcations and on a plane without curvatures. Early modern textbooks of logic often included bewildering diagrams showing the spatial array of logical relations as rectangles with any number of connective links.22
Ong insists that this linear charting of intellectual motion was deeply indebted to the invention of the printing press, and specifically to the rectangular uniformity of its page and its type. The rectangle of the printed page provided a coordinate system in which geometrical analysis and speculation on the extent of linearity and calculability could take place. The emerging systems of natural history sought to capture the variety of natural forms in catalogs that showed linear dependence of species very much like the diagrams of early modern logicians.23 Works like Luca Pacioli’s De divina proportione (1509) sought to arrive at a universal, geometrically modular typeface that in turn would be able to represent a universal language, actively sought by European learned societies at the time. Pacioli was equally convinced that the human face exhibited geometric proportions; neither type nor face was as yet subject to the kind of intuitive physiognomies that in the late eighteenth century would brush away all geometric and linear constructions.24
In the notion of proportion, however, the other, “Platonic” side of the new geometry came to the fore: proportion was “divine” insofar as it could not be assigned an exact number, yet it was an integral part of geometric patterns and, what is more, a sign of beauty. The circle and the sphere in particular embodied this rest of divinity in a world that was increasingly defined by numerical values. The relation between circle and square (and their relation to the human body, as in Leonardo’s Vitruvian Man), the relation between the circumference and the diameter of the circle, and of course the golden ratio were favorite objects of speculation in the Renaissance, as were the Platonic regular solids and their relation to the sphere. Indeed, the ontological 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
