The Cylinder. Helmut Müller-Sievers

“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 and one prism pair. These cylinder chains transmitted and converted motion across the plane of the machine from the motor to the tool; they followed the same “phoronomic” laws as their elements and were fully determined (even though describing them mathematically remained difficult).

      Mechanisms that employed the three cylindrical pairs were at once the basis and the ideal of Reuleaux’s kinematics because they excluded all interference by outside (in Reuleaux’s terms, “cosmic”) forces and thus allowed for a coherent logic of machine elements. By calling his pairs Umschlusspaare and their combination “chains” (Ketten), Reuleaux invoked an embodied logic of material elements—Kettenschluss is, after all, the German word for syllogism.⁴¹ The overall goal of Theoretische Kinematik was “kinematic synthesis”—which, in the wake of Kant’s distinction between analysis and synthesis and with a view of making good on Monge’s and Ampère’s program, Reuleaux conceived as the science of deducing kinematic assemblages a priori, regardless of material or even of purpose.⁴² Reuleaux coined a word to invoke both the exclusion of cosmic forces and the a priori necessity of kinematic design: zwang(s)läufig. It has since entered the German vernacular with the meaning “inevitable”; Kennedy translates it as “constrained,” and Reuleaux in a note offers the Greek “desmodromic,” which has caught on in certain engineering circles.⁴³

      Reuleaux was too much of a practitioner not to know that many mechanical linkages cannot be converted into cylinder chains with fully constrained pairs—ropes and belts and springs, for example, could not be enclosed, and the strain on the material in enclosed links often exceeded the metallurgical capacities of his time. Nonetheless, he understood the history of machine design to be a logical—a zwangsläufig—development from “force-closure” to “pair-closure.” Force-closure, like the link between a cam lobe and a valve or between the wheel of a locomotive and the rail, is open to “cosmic” interference (valve float or wheel slip); pair-closure—its basic forms being embodied in the three cylindrical enclosed pairs—eliminates such interferences by systematically forcing motion in one direction to the exclusion of all others. The change from one to the other provides, according to Reuleaux, a parameter by which to measure progress in machine design: “The question now arises:—what is the special kinematic meaning or nature of the changes by which the machine has been advanced to its present degree of completeness? . . . I believe the answer to this question is:—the line of progress is indicated in the manner of using force-closure, or more particularly, in the substitution of pair-closure, and the closure of the kinematic chain obtained by it, for force-closure.”⁴⁴ One way of describing this development in kinematic terms—and in terms provocatively contrary to liberal philosophies of history—is to chart it as the successive elimination of freedoms. For engineers, an object within three-dimensional Euclidean space has six degrees of freedom: it can move along the three axes of space and it can rotate around them. The motions of mechanisms (as opposed