Читать книгу The Cylinder - Helmut Müller-Sievers - Страница 10

CHAPTER 2

The Rise of Kinematics

Kinematics is the science of forced motion, of motion in mechanisms and machines. Interest in such motion emerged once the concern with the origin of motion and the nature of motive forces moved from the domain of metaphysics to the newly energized discipline of physics. The steam engine diminished the eighteenth century’s fixation on origins—debated in the innumerable Academy prize questions about the origins of motion, of species, of language, of ideas, of property—because it normalized the generation and harnessing of motion and because it focused attention on the measurability of relations between previously unconnected phenomena. Heat ceased to be a separate substance with separate properties; it became an effect of motion, but it was this effect in such a way that nothing was lost or unaccounted for in the transition from one state to another. The old distinction between cause and effect whereby the former was inferred from the latter in a metaphysical leap—there must be forces because there is change in motion—was superseded by strict equivalencies between contiguous phenomena—there is heat whenever there is motion, and vice versa: causa aequat effectum.¹ The actio in distans that governed, and bedeviled, Newton’s mechanics gave way to processes that were conceived as contiguous or, as historians later would call it, analog.² Insofar as it translated every physical change into a fully measurable effect, the steam engine was, on a very fundamental level, nothing but a mechanism to transmit the “motive power of heat” present in the universe. Kinematicists, who conceptualized and facilitated this transmission, could therefore claim to be concerned with the very essence of machines.

What engineers had tacitly presupposed since the middle of the eighteenth century found its basic expression in the first law of thermodynamics: not only are forces convertible into one another, but such conversions also happen without absolute loss. The accounts of all transactions in nature are always balanced, nothing is added to and nothing lost from the overall sum. The law of the conservation of force confirmed that machines were calculable and therefore scientific objects and that the input in energy equaled the output in work minus the inevitable price paid to the environment in terms of evaporation, cooling, friction, and so on.³ Their calculability as material systems in interaction with their environment distinguished nineteenth-century machines from the automata of the eighteenth century, which needed to be insulated against their environment, had an input consisting in some form of kinetic energy, and had an output that was not measurable—or, in the case of pendulums, clocks, and other instruments, was measurement itself.⁴

The interconversion and the conservation of force—as well as the interconversion of the knowledge of engineers, physicists, and physiologists—provided a finite frame in which translatability was a much more concrete and immediate concern than originality.⁵ Of course, the question of the origin of force was not “solved” by this approach, but, in a fashion not untypical of the epoch’s concentration on practicability, it was pushed toward the margins of metaphysics and religion. Under the auspices of the laws of thermodynamics, the earth, fueled by the heat of the sun, became part of a vast cosmic heat engine; the individual machines built on earth did nothing but intercept and utilize the stream of energy flowing through them. Cosmic heat, stored in subterranean coalfields, needed only to be reignited to provide energy for untold machines independently of their location.

Unfortunately, physicists also discovered that while forces could be translated into one another, the overall flow of energy was unidirectional and irreversible, from hot to cold. This second law of thermodynamics marked the beginning of ecological thinking; remarkably early, physicists and philosophers realized that human beings were, in the words of a French observer, “nothing but concessionaires” of the earth’s finite resources.⁶ Physicists, many of them devout Christians, scrambled to reconcile this inglorious dissipation with the biblical apocalypse, but the fact remains that ecological thinking is characterized by a renunciation of transcendence and divine intervention: while in Newton’s universe God still had to intervene to correct potentially catastrophic irregularities of planetary motion, the fully analog and slowly stalling universe of the nineteenth century no longer had an opening for such correction. Not the origin of forces but their end became a major preoccupation for the epoch; entropy, the inevitable descent of all organization into undifferentiated matter and meaningless noise, was the flipside of the fully calculable universe.⁷

In this situation, kinematics as the science of mechanical energy exchange and transmission rose quietly to prominence, not only because under the first law of thermodynamics every machine is a transmission mechanism anyway, but because under the second law the transmission is that part of a machine that can minimize entropy—by finding the best paths, by reducing stress on materials, and by avoiding as much as possible leakage through friction.⁸ This is not its stated goal, and the first great theorist of the field, the German Franz Reuleaux, explicitly excluded all material considerations; but in the end he too dreamt of a totally negentropic machine, one that would run in perfect silence with the least amount of energy loss.

Modern kinematics owes its theoretical formulation and its formation as a discipline to the emergence of new schools and curricula, particularly in Napoleonic and post-Napoleonic France.⁹ While France—to say nothing of Germany—lagged behind in the development and industrial deployment of steam engines, the Grands Écoles, founded in the wake of the French Revolution, were among the first institutions to reward engineers with academic positions and to urge mathematicians to think about the practical implications of mechanisms. Already in 1794, Gaspard Monge proposed courses on the theory of machines that would focus on the elementary mechanisms of force transmission: “By these elements are to be understood the means by which the directions of motion are changed; those by which progressive motion in a right line, rotative motion and reciprocating motion, are made each to reproduce the others. The most complicated machines being merely the result of a combination of some of these elements, it is necessary that a complete enumeration of them should be drawn up.”¹⁰ The mathematician Monge here identifies machine transmissions as instantiations of Leonard Euler’s earlier observation that the motion of all rigid bodies may be broken down into translation along a straight line and rotation around an axis.¹¹ To transfer motion to act at any point in space and to act as translational, reciprocal, or rotational motion, the machine designer has to devise a chain of joints and linkages that best embodies and combines motions. Franz Reuleaux will later summarily call these chains cylinder chains.

The recognition of rotation as an irreducible and entirely technical form of motion meant that the moving object under investigation and construction could no longer be conceived as a mathematical point; points, lacking extension, cannot rotate. As Kleist’s choreographic criticism had implied, the coherence and predictive success of Newton’s mass point mechanics were in large part predicated on the fact that celestial objects in motion could be reduced to geometrical points because they were so far away and their orbits were so large; at close range and in rapid repetition, however, otherwise negligible imperfections (in the axial symmetry of an object, for example) were magnified and could quickly lead to the breakdown of a system of linkages. This is why standardization and precision tool making would take the place of mathematical solutions in nineteenth-century engineering.

Newton’s celestial objects were moving along straight-line paths, or on paths that could be analyzed as the result of forces jointly impacting an imaginary center where all mass was concentrated. Rotary motion, by contrast, had to be conceived as the impact of two forces at separate points of an extended body. To reiterate, it makes no sense to speak of the rotation of a point, but neither does it make sense to speak of a single rotating force.¹² Louis Poinsot, also a product of the new French education system, argued that rotation should be viewed as the result of a “couple” of forces, acting equally from opposite directions on a line drawn through the center of a rotating body. Thus rotation can be quantified as the product of the forces times the length of the line on which they act: this is the measure of torque—a quantity unknown to the eighteenth century—which even today is the true measure of the output of machines, most prominently the automobile engine.¹³ (A good example is turning a car’s steering wheel: one hand pulls downward, the other pushes upward, and both are at an equal distance from the center of the wheel. Before the introduction of power steering, the diameter of steering wheels in heavy trucks was particularly large to help the driver expend less force in turning the vehicle.)

Kinematics relies on a still more restricted description of motion than that outlined by Newton and amplified by Poinsot. It is defined as a view of motion independent from the forces causing it. Since machines are, from one point of view, manifest attempts to eliminate random, or, as Franz Reuleaux would say, “cosmic,” forces, kinematics is always “kinematics of machinery.”¹⁴ The text that is most often mentioned as the declaration of independence of kinematics, André-Marie Ampère’s Essai sur la philosophie des sciences (1834), clearly recognizes this interdependence of machinery and geometric description:

It [i.e., the new science of kinematics] should treat in the first place of spaces passed over, and of times employed in different motions, and of the determination of velocities according to the different relations which may exist between those spaces and times. Furthermore it should study the various instruments by means of which one motion can be changed into another; so that if one conceives of these instruments as machines (as is usually the case) one must define a machine not, as one customarily does, as an instrument by means of which one can change the direction and the intensity of a given force, but as an instrument by means of which one can change the direction and the speed of a given motion.¹⁵

While French theorists put serious efforts into founding and institutionalizing kinematics as a deductive science, British engineers were attacking its practical problems. Kinematically speaking, the rise of the steam engine as the motor of the Industrial Revolution was the result of a specific mechanism to “change the direction and speed of a given motion,” more precisely the (reciprocating) translational motion of the piston, into the rotational motion of the working beam. It was invented by James Watt in 1784 and was immediately patented so that it could reach the open market at the very beginning of the nineteenth century. Only then could the proliferation of cylindrical machines in the nineteenth century really begin.¹⁶

This mechanism, commonly called “Watt’s parallel motion,” changed the steam engine from a pendulum into a fully rigid mechanism. It connected the piston that rose from and was pushed into the cylinder with the beam that pivoted on a central column.

The pivoting beam was part of the early architecture of steam engines, which were primarily used to pump water. Before Watt, only the downward stroke of the engine was powered: either the rapid cooling of the steam under the piston created a vacuum that pulled the piston down, or steam was injected above the piston. In this configuration, where the piston pulled on the beam (and the beam pulled on the pumping vessel), it was enough to use chains or ropes as a connection; they were run across the ends of the beam, and the kinematic conflict between the semicircular motion of the beam and the straight motion of the piston was reconciled—just as it was in Kleist’s marionettes—by the slackness in the connection (fig. 1).

FIGURE 1. Watt’s 1774 engine. The piston (and the valve gear) are connected to the beam by a chain; the power stroke can only be downward. Reprinted from Thurston (1902, 98).

This paradigm had to be changed when Watt began to power both the up- and the downstroke of the piston by using steam as a positive (expanding) rather than as a negative medium. Now the piston was pushing up on the beam as well as pulling it down; ropes and chains did no longer work, and a simple rigid rod without a mediating mechanism would have destroyed the machines in a very short time—if the piston were pushed along a line that deviated from the cylinder’s axis, it would scrape against the inside walls, destroying its symmetry and losing the ability to seal and maintain pressure.¹⁷ Even without these difficulties, the practical problems of boring or casting accurate enough cylinders in sufficiently strong materials and of finding lubricants to minimize the inevitable friction proved very hard to overcome for most machine builders in the late eighteenth and the very early nineteenth centuries.¹⁸ One of Watt’s many advantages in the race for efficient engines was that through his partner Matthew Boulton he could intervene directly in the manufacturing of cylinders, asking for more precision in boring and for stronger alloys.¹⁹ To Boulton he first announced his discovery that a rigid linkage configured the right way could guide both the up-and the downstroke of a double-acting steam engine without stressing the materials involved.

In a formulation at once revelatory of the truly empirical process of engineering and of the stunning novelty of motion conversion, Watt wrote of the contraption he called “parallel motion”: “When I saw it work for the first time, I felt truly all the pleasure of novelty, as if I was examining the invention of another man.”²⁰ Yet like so many engineering advances in the nineteenth century, parallel motion was an avoidance of conflict rather than an invention of something entirely new. The mechanism simply caught two semicircular movements at the point where they intersected along a seemingly straight path (fig. 2). One was the movement of the beam O_A—A, which in kinematic nomenclature was called the crank (the Kurbel, of which Kleist’s Herr C. dreamed); the other was the link O_B—B affixed to an opposite wall, called the follower. Both were connected by a third link A—B, the coupler. As the crank moved up and down, it led the follower into a mirror image of its own motion whereby a point M on the coupler was forced to trace out an elongated figure eight, the sign of infinity. If the proportions of the links were chosen appropriately and the movement of the crank was restricted accordingly, M traced a line that was approximately parallel to the beam’s support column. A piston rod, attached to C, could push and pull in a line extending from the cylinder’s axis. Depending on the machine’s architecture and size, Watt translated this parallel motion horizontally by means of pantographs—linkages based on the parallelograms that had long been used to translate writing and drawing across a plane—which yielded other parallel points M′ able to drive a valve train or an auxiliary pump (fig. 3).²¹

Watt’s mechanism not only allowed for a potentially infinite increase in power output but also universalized the use of steam engines just as much as fossil fuel rendered them independent of natural location. The four-bar linkage (the hatched line at O_B and O_A on the left of figure 2 indicates a fixed frame and counts as one bar, just like the “floor and datum” on the right) is the most economical way of mediating between translational and rotational motion. To repeat, such mediation is necessary because in a finite mechanism (unlike in the universe or in a gun) every translational motion needs to be “returned,” every straight motion needs to be reciprocal or oscillating.²² Using variants of the four-bar linkage, engineers could eliminate the working beam and configure machines for a hitherto unimagined variety of purposes—or else utilize the transmission as the machine’s tool, as is the case in motorized vehicles. The slider-crank mechanism—an avatar of the four-bar linkage—became the most successful of these linkages: first in the locomotive, then in the internal combustion engine, it allowed the motor to produce nothing but rotation.

FIGURE 2. (a) Watt’s “parallel-motion” linkage in schematic form: O_A—A is the beam’s arm, acting as a crank; A—B is the coupler; O_B—B is the follower, anchored to a wall. The point M on the coupler will trace out a figure eight, part of which is “straight” and can be used to guide the piston rod. (b) The mechanism on Watt’s engine. Point M is transposed to M′ by means of a pantograph and there guides the rod of a pump. The sun-and-planet gear on the working side of the beam would be useless without the continuous motion provided by the parallel linkage. “Floor and datum” is what Reuleaux (and Heidegger) call Gestell. Reproduced with permission of The McGraw-Hill Companies from Richard Hartenberg, Kinematic Synthesis of Linkages,© 1964.

From a kinematic point of view, it is irrelevant where the motion of a mechanism originates and where it is utilized, as kinematics is not concerned with forces or with stresses on material that might result from the impact of forces.²³ Kinematic transmission functions without a fixed origin (such as straight-line motion) and without a determined destination (such as pure rotation) but is concerned (to use Walter Benjamin’s term) with translatability (Übersetzbarkeit) as such. All that linkages, and machines in general, need is a frame that determines the orientation of their movements; it generally consists in anchoring one link to an immobile part that in figure 2 is called the “floor and datum” but that in Reuleaux’s seminal terminology becomes Gestell.²⁴

FIGURE 3. Drawing of a Watt and Boulton steam engine, after 1784. The parallel-motion linkage is on the right; on the left is the sun-and-planet gear driving a flywheel. Also visible on the left side is a governor, another of Watt’s inventions. It rotates with the engine stroke and shuts down the steam supply if the machine runs too fast. © Science Museum / Science & Society Picture Library—All rights reserved.

FIGURE 4. Two of Reuleaux’s teaching models. The curve traced by a point on the coupler depends on the length of the links, and on which of them is immobilized by the Gestell. Reprinted from Reuleaux (1876, 68, 71).

The curves traced by a point on the coupler (the link d—e in the linkage on the left in fig. 4) are an instructive example of the irreducible empiricism and pragmatism in the construction of kinematic transmissions. They change in proportion to the length of the individual links and to the position of the Gestell, but the rate of this change and the bewildering variety of the resulting curves defeat attempts to describe them algebraically or in any other form of abstraction. This was true at least as long as the means of representing these curves were, like the mechanisms that produced them, analog; computer programs now can easily model coupler point curves, and the problem, like many others, has disappeared from the problem sets of kinematics students. Franz Reuleaux felt that the best way to teach the properties of four-bar (and other) linkages was to build (and license) an extensive collection of teaching models, which can still be admired, for example, in Cornell’s Sibley School of Mechanical and Aerospace Engineering. Seeing these models in motion—or seeing Theo Jansen’s fantastically inventive linkage “beasts” prowl the beaches in Holland—gives us a rare sight of kinematics liberated from the servitude to motor and tool.²⁵ They show that there is distinct grace and beauty in forced motion, as Kleist’s Herr C. claimed with seeming contrariness. Uncovering the aesthetics of forced motion as an object of contemplation, as a driving force in mechanical engineering, and as an element in nineteenth-century literary culture is a goal of the following pages.

Such a goal was far from the mind of Franz Reuleaux, the great German synthesizer of machine design and kinematics; with his Theoretische Kinematik of 1875 he wanted to provide a space for kinematics on the curriculum of German research universities, which had been founded by men around Friedrich Schiller for whom all things mechanical were anathema. While experimental physiologists, despite operating with rather gruesome empirical remainders themselves, had managed to secure for themselves a prestigious place in the German research university, mechanical engineering was still relegated to professional schools and para-academic institutions. Reuleaux, who had traveled widely in Europe and in the United States, felt that German engineering products stood no chance in an increasingly globalized market and that it would behoove the Second Reich to centralize engineering training and raise it to a par with other academic disciplines.²⁶

In the German context, any discipline wanting to graduate to a full-fledged science had to meet two fundamental requirements: it had to be in discursive control of its own principles and presuppositions, and it had to be able to give a coherent account of its own history. In the case of experimental physiology, for example, this meant that the dubious principle of Lebenskraft (vital force) had to be abandoned in favor of the first law of thermodynamics and that a careful rewriting of its history, especially with regard to Romantic visions of vitality (including Goethe’s), would integrate physiology into the context of German intellectual history. Many of Herrmann von Helmholtz’s popular lectures were devoted to this task.²⁷ In the case of mechanical engineering this meant that all contingent factors in machine design—such as the metallurgy of machine parts, the turbulences of power generation, the economic concerns of the manufacturing process, the social conditions of factory workers—would have to be bracketed, and the logic of machines developed deductively. Relying on the definitions by Ampère and other theorists, Reuleaux realized that an a priori deduction of the logic of machines could proceed only from the kinematics of machinery. The Theoretische Kinematik (translated into English in 1876 as Kinematics of Machinery) seeks to unfold this logic beginning with the most fundamental givens of material contact, and it invents a symbolic language in which machine elements can be classified and their combination be taught. At the same time—hidden in the vast body of his book—Reuleaux sketched a history of machines and mechanisms that emulated in scope the grand historico-philosophical designs of German historicism.²⁸

With a good measure of irony, though not without systematic pride, Reuleaux reached back to the pre-Socratic sage Heraclitus for his most fundamental statement: “Everything rolls.”²⁹ Everything in a machine is in contact with everything else in a motion that is at the same time rotational and translational. Motion in and of machines is always relative motion (anchored by the Gestell of its frame), and the successive positions of one extended body in relation to another can always be configured as one curve rolling off another. In the part entitled—with obvious reference to the opening chapter of Immanuel Kant’s Metaphysical Foundations of Natural Science—“Phoronomic Propositions,” Reuleaux demonstrates this relationship first as that between a moving and a fixed line. The successive positions of the moving line P—Q (or of any other figure through which a line can be drawn) with respect to the line A—B can be described by two separate lines: first, as the line between the successive points around which the line rotates (its poles) as it moves along the x axis in an imaginary Cartesian coordinate system (the line O₁, O₂, O₃ in the following illustration); and second, as the line between the successive points that indicate the rate of rotation along the y axis (the line M₁, M₂, M₃) (fig. 5).

Contracted from polygons into smooth curves, these curves fully describe the instantaneous position 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 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 to those of a ship or a plane, from which many of the technical terms for the degrees of freedom are taken) are constrained to one plane, as in Watt’s parallel mechanism, thus eliminating all freedoms except rotation around an axis and translation along it. These two freedoms, as well as their combination in the motion of a screw, are embodied, as Reuleaux had casually remarked, in the body of the cylinder.

Against this backdrop, the history of machine development, which Reuleaux inserted as a compact chapter into his Theoretische Kinematik and later dispersed over the second volume of the Kinematik, which appeared in 1900, amounts to a history of the progressive elimination of “cosmic” freedom.⁴⁵ “We have recognized and examined in certain pairs of kinematic elements the property of force-closure, by which a certain amount of kosmic freedom is left in the machinal system, and seen that it has been for thousands of years the aim of invention to limit or destroy this freedom.”⁴⁶ Reuleaux’s Ur-machine is a single cylinder: the fire drill, a pointed stick twirled by hands on a wooden cavity with the purpose of igniting the wood itself or fibers placed around it. So long as human hands twirl the stick, there is pair-closure only between the recess and the point of the stick. The next step consists in replacing the hands by a rope, which does not alter the nature of the closure but speeds up the rotation. Then a stone or a fitted piece of wood is placed on top of the rotating stick in such a way that all motion except rotation is eliminated. Now the twirling stick is part of a pair-closed chain that produces fire in a fully predictable manner.

A more contemporary but perhaps not equally felicitous example is the steam locomotive. It replaced the horse-carriage, which had been improved upon in various ways, for example in shock absorption and in the development of steering gear, but which was still beset by the potential disturbance of cosmic forces, such as uneven roads or drunken coachmen.

Force-closure still remained, if nowhere else at least in the preservation of the direction of motion, which still demanded accustomed animals and an intelligent driver. Men naturally attempted to replace this force-closure by pair-closure. In the Railway the rails are paired with the wheels,—force-closure is used only to neutralize vertical disturbing forces. The step thus made in the direction of machinal completeness . . . was in reality no other than the uniting of the carriage and the road into a machine. The rail forms a part of this machine, it is the fixed element of the kinematic chain of which the mechanism really exists. . . . In opposition to this we have the problem of steam locomotion on common roads, which has been so feverishly taken up again within the last few years, but the solutions of which seem doomed to eternal incompleteness, for they are self-contradictory. It is desired to make something which shall be a machine, but in which at the same time the special characteristic of the machine—the pairing of elements—may be disregarded.⁴⁷

To be sure, the pair-closure between the locomotive and the rail is only approximate: it is achieved by the weight of the engine (and in fact often breaks when the train has to climb a steep incline). What Reuleaux means by the inner contradictoriness of the automobile is that the wheels of the car cannot form a pair-closure with the road if the automobile is defined as a vehicle that can go anywhere by itself; he mentions the recent discovery of wheels made of “India-rubber,” which try to emulate rails insofar as “vulcanized India-rubber, externally flattened upon the road, serves as a smooth uniform surface for the rigid tread to run upon, thus corresponding generally to the rail of the railway”;⁴⁸ kinematically speaking, however, the automobile is a failure because the pair-closure of its engine (the slider-crank-linkage) is stunted by the weak force-closure of its contact to the road. The further development of rubber wheels and the improvement of roads by means of another cylindrical machine, the steamroller, will alleviate this weakness, but every instance when a car spins its wheels or swerves off the road or just out of its lane is a testimony to the justness of Reuleaux’s observation.

Although in the use of his terminology Reuleaux seemed to emulate Kant’s critical philosophy, his view on the history of machines was Hegelian. Very much in the tradition of Hegel, Reuleaux tried to understand the history of machines and mechanisms as a slow but logically driven and often dialectical process toward maximum efficiency. His ideal was a machine, consisting of absolutely rigid elements connected by cylindrical pair-closures, that would capture and convert the energy flowing through the cosmos with as little noise and as little loss as possible. But this historical dialectic was the limit of his Hegelian leanings; in cosmological terms Reuleaux was thoroughly modern. Like Poinsot, like Auguste Comte, and like the foremost physicists of his time, he conceived of the cosmos, not as a living being (as Hegel still did), but as a vast machine driven by heat, in which the planets were the remnants of a linked planar mechanism. Perfecting transmissions, from this perspective, meant combating entropy in the only arena possible, namely by slowing down the dissipation of energy in fully linked, “pair-closed” machines.

Reuleaux also paid attention to the devaluation of human work. Like most engineers and scientists in the latter half of the nineteenth century, he was keenly aware of the destructive and dehumanizing potential of industrial modes of production and sought to confront the “burning question of our time, the question of the worker,” with proposals of his own.⁴⁹ Characteristically, he saw the problem in the motor end of the machine: it is the logic of capital, he argued, that requires ever more powerful motors, which in turn lead to larger factories and more alienated labor. His own solution proposed smaller, yet kinematically efficient machines that would need fewer, perhaps even just one worker to attend them—automobiles, as it were, that did not move. We will see in chapter 5 how Karl Marx, unconcerned with the kinematic implications of factory work, shifted the discussion almost exclusively to the tool end of the machine.

Far more startling than the faith Reuleaux placed in the desmodromic progress of mechanization is the fact that he did not reflect on the shape that dominated every level of his investigation: the cylinder. We have seen that on the abstract level of phoronomy he conceived of relative motion as cylindrical rolling; on the elemental level of kinematic pairs he identified the cylindrical screw and its extremes as irreducible connectors; on the level of mechanical assemblies he developed a grammar of cylinder chains; and on the grand historical scale he began with an Ur-cylinder (the fire drill) and then described mechanical progress as the replacement of contiguous by cylindrical closures. Yet nowhere did Reuleaux look beyond the confines of kinematics and identify other cylindrical structures, such as rolling mills, the pneumatic tube delivery, or the tin can; nor did he ask why this shape, rather than any other, so dominated the machines and, as we will see, the culture of his epoch. This oversight is partly due to the natural myopia of the immersed witness and practitioner, but partly to the effort it takes to see that motions, and the shapes through which they are transmitted, are historically and culturally identifiable phenomena. Our view is traditionally trained on the motor or on the tool, not on shape-dependent transmission. The following chapter will begin to right this oversight by adding historical depth to shapes and motions.

The kinematic epoch that began so neatly in 1800 with the expiration of Watt’s patent for parallel motion came to an end somewhere between the large-scale use of electrical motors, the discovery of radio transmission and X-rays toward the end of the nineteenth century, and the conflagrations of World War I. It was based on the visible, “analog” contact between moving parts and, more particularly, on the taming and conversion of rotation and translation. All of this was possible because with the emergence of the steam engine the kinematic problem of forcing and converting motion could be detached from concerns over the generation of power. The early French theorists of kinematics held out the possibility of devising a meaningful geometry of machine motion that would allow the construction of machines entirely on the drawing board. The experience of British machine builders showed that everywhere in the development of machines empirical factors would trump theoretical insight, in particular when the demands of the market and necessities of exploiting natural resources came into in play. Reuleaux, finally, sought to integrate practical and pedagogical concerns, but he also hoped that a grammar of forcing motion could be constructed that would allow the generation, the “synthesis,” of transmissions, and with it the construction of machines for any purpose whatsoever. The unthought element in this entire development was the cylinder.