of the translating and rotating line P—Q in relation to the line A—B, which lies on the same plane (it is “con-plane”). Reuleaux calls these curves Polbahnen; his translator Kennedy calls them centroids (later changed to “centrodes”).30 The purpose of this abstraction is to show that the relative planar motion of any two bodies can be fully described once their centrodes are known, and that this relative motion can be described as a rolling.31
Machine parts, then, just make actual what is potential in any relative motion of two rigid bodies in a plane. Reuleaux operates with the abstraction of moving points and lines only because he strives for maximum generality—for the justification of his law that everything rolls. He is fully aware, of course, that the subject of kinematics is machinery: that is, an assembly of rigid bodies that have additional properties, even if one abstracts from material and from the forces to which they are subject.32 The reciprocal rolling of the centrodes, as soon as it is conceived as being performed by two extended bodies moving in the same plane, must be understood as the rolling of one cylinder against another, for it is the cylinder alone that has an extended curved surface and a fixed axis of rotation.33 Even if one of the bodies does not move, the other can roll on it, as a locomotive’s wheel rolls on its rail (which is conceived as a cylinder with infinitely large diameter). The application of the Heraclitean law of rolling to the real world of extended machine parts therefore reads: “We may extend the law just enunciated for plane figures equally to the relative motion of solids . . . : Every relative motion of two con-plane bodies may be considered to be a cylindric [sic] rolling, and the motions of any points in them may be determined so soon as their cylinders of instantaneous axes are known.”34
FIGURE 5. The relative translation and rotation of an extended body represented as the rolling of one body (P—Q) off another (A—B). Reprinted from Reuleaux (1876, 62).
Even though the cylinder as an embodied motion is crucial for the understanding of the relative motion of extended bodies, Reuleaux introduces it in the first part of his theoretical kinematics without further comment or reflection. Far-reaching consequences of this conception could be explored: for example, the oscillation of rolling as an intransitive verb of motion and as a transitive verb denoting perhaps the most important industrial processes of the nineteenth century. Spheres, for example, can roll on one another (as they do in ball bearings), but only cylinders can roll something. Yet the cylinder, although everywhere present, is neither thematized nor generalized by Reuleaux.
It is worth reflecting for a moment on Reuleaux’s “discovery” of centrodes, because it repeats on a higher level of generality the epochal shift from pendulum to crank that we have seen playfully discussed in Kleist’s story about the marionette theater. Centrodes belong to a class of curves known as “cycloids,” which are traced out by a point rolling on a circle, either on its periphery, or its interior, or outside its periphery as long as it is rigidly linked. The most prominent and universally visible example of such an (interior) curve in the nineteenth century was undoubtedly the motion of the crosshead on a locomotive wheel, which Heidegger rightly counted among the essentially technical motions.35 But cycloids were of equally great importance for premodern astronomy, where the motion of the planets was conceived as their rolling on the surface of celestial spheres, and the apparent irregularities in their orbit were explained as epicycloids—as rotation upon a rotation that might look from the center of the system like a slowing down or an acceleration. Doing away with this extremely complex system and replacing it with the comparative simplicity of the earth’s eccentric position and with gravitational forces acting instantaneously across the void had been Copernicus’s and Newton’s great innovation. The return of the cycloid in the nineteenth century, then, was a return of ancient celestial mechanics in the shape of machines and mechanisms—a return of a concept of cosmic grace and of cosmic coherence that characterized the newly closed system of thermodynamics.
The drama of this epochal difference was played out in the delicate frame of the pendulum clock. Galileo had initially thought that the period of the pendulum’s swing was isochronous—that it would mark identical time intervals if all outside factors like friction were eliminated. Huygens famously proved this assumption wrong and showed instead that only if the pendulum was forced by an outside constraint (like a metal “cheek” on each side of the swing) to follow the line of a cycloid rather than that of a circle did it really count equal intervals. For Reuleaux, this episode strikingly exemplified the difference between theoretical geometry—descriptively accurate but practically worthless—and the theory of constrained motion (Zwanglauftheorie) that his Kinematik proposed to unfold.36 This is the kinematic reason why Reuleaux, and many machine theorists with him, understood machines to be part of the cosmos, not artifacts alien to it.
Reuleaux also remarked explicitly that rolling always meant the rolling of one body on the surface of another.37 That is, already on the most general level of his system, he conceived of kinematic phenomena as relations of pairs. This admission of an “original duplicity” differentiated the empirical approach of engineers from that of philosophers and theologians, who were committed to the search for first and singular causes. Reuleaux did not reflect on this stance; but he did carry it over into the second of his major contributions to the science of kinematics, the concept of kinematic pairs. If every motion in a machine was relative, Reuleaux argued, it could be conceived as the contact motion of one part against another. Therefore, the smallest element of a machine was a pair or couple (just as the smallest element in Poinsot’s theory of rotation was a couple of forces). These couples, like the linkages on their plinths, had to fulfill certain conditions—one of their elements had to be the other’s Gestell, the fixed element had to follow the form of the mobile element, and the joining had to exclude all other motions (“freedoms”) except the one that was desired. The ideal couples to meet all of these conditions were the ones where one element fully enclosed the other—Reuleaux called them Umschlusspaare or enclosed pairs.38
The three elementary enclosed pairs Reuleaux deduced were by necessity all cylindrical. For when one body enclosed another and still needed to move, it could slide along the enclosed body’s axis, rotate around it, or, ideally, do both. The three kinematic couples, then, were the revolute joint, the prism, and the screw-nut couple (fig. 6).
In a way that would become important when screw theory at the end of the nineteenth century generalized the motions of rigid bodies, these could be understood as versions of the screw: the revolute pair as a screw-nut pair with a thread tending toward zero, the prism as a screw-nut tending toward infinity. These three links exhausted the possibilities of enclosed pairs, since in planar motion—motion across a precise plane as is necessary in machines—no other motions than sliding, rotating, and their combination are possible. Indeed, “all three are well known in machine construction,—the screw pair both in fastenings and in moving pieces; the pair of revolutes in journals, bearings, &c. and the prism-pair in guides of all sorts.”39
FIGURE 6. The three kinematic couples, from left to right: the revolute joint (which contains rotation, as in a wheel hub), the prism (which contains translation, as in a guide rail), and the screw and its nut. Reprinted from Reuleaux (1876, 43).
These couples by themselves did not yet have a determinate use; they were like the roots of words that were not yet inflected and connected to meaningful sentences. The next larger units therefore were kinematic chains—mechanisms in which cylindrical pairs served as joints. Watt’s parallel linkage was such a Zylinderkette,40 since it—like every four-bar linkage—consisted of four revolute pairs connected by rigid links; the slider-crank mechanism typically consisted of two revolute pairs
