I’m reading Reuleaux’s 1874 book on machine kinematics, or rather its English translation, and I thought I’d make some notes. Reuleaux formulated kinematics as it is studied today, in terms of lower and higher kinematic pairs with varying degrees of freedom, such as revolute, prismatic, and cylindrical joints.
This is one of the rare cases where the Google scan is of less bad quality than the Archive’s own scan; kinematicsofmach00reulrich.pdf has the left side of many pages cut off, while so far Google’s kinematicsmachi01reulgoog.pdf seems okay. As always it is lower in resolution and color depth, but so far this seems to be an improvement, accelerating as it does the rendering. It seems to lack OCR text, though, while the Archive’s own scan has somewhat passable OCR. The OCR is still not good enough to copy and paste, but sometimes it may be faster to edit the results than to retype from scratch.
The Archive’s scan also does a less appallingly bad job of capturing the figures, though it still leaves something to be desired. (Compare p. 31, for example; 49 of 646 in the Archive’s scan, 52 of 651 in Google’s.)
Mupdf in particular handles the total absence of OCR text rather disappointingly when you initiate a search, freezing for a few minutes before it comes to the end of the document.
Many aspects of the typography and usage of this work deviate from modern standards, but none more glaringly than its use of increased letter spacing for emphasis, where modern practice, or even common 19th-century English practice, would use italic. This was of course necessary in German blackletter typesetting, which entirely lacks italic, but I suspect that even the original German version of this work was set in modern humanist letters rather than blackletter. (Perhaps I should go back and check.) I often fail to notice this on the first reading of a passage.
Where I’ve quoted passages using this form of emphasis below, I’ve replaced it with italics.
The typography does use italics to distinguish mathematical variables and references to points in figures.
It’s interesting that Reuleaux traces the development of kinematics though a whole series of 19th-century researchers, including (to my surprise) Ampère. I had sort of thought that the systematic study of kinematics had sort of been stuck at the level of Archimedes’ “simple machines” still taught in the vulgar elementary schools: the inclined plane, the screw, the lever, the wheel and axle, the pulley, and the wedge. But indeed Reuleaux outlines the development of the ideas by Galileo, by Hâchette, by Borgnis, by Lanz, by Monge, by Willis, by Ampère, by Newcomen, by Watt, and by a dozen others, prior to himself. Justifiably, though he characterizes all their theories as “wrecked”.
It’s intriguing to see the etymology and sense development of “automatic” and “automation”:
Moreover, as a further fruit of this uncertainty there has been an attempt to construct yet another special study which demands mention. This is the so-called Automatics, the study of the realization in mechanism of motions either supposed or given by mathematical expressions. For this further attempt at separation we have to thank the engineer E. Stamm, who wishes again to divide his subject into pure and applied parts, as fully described in his Essai sur l’automatique pure, 1863.
Of course the term “automaton” is much older.
When Reuleaux speaks of “the real end before us — the progressive development of the machine,” it is difficult to avoid being reminded of Forster’s The Machine Stops from a few years later.
The modern development of flexure design methods is a disquieting echo of the disjointed and “wrecked” development of mechanisms of which Reuleaux complains; the stamp-collecting botany of the Handbook of Compliant Mechanisms closely resembles what he deplores in Monge’s and Lanz’s work. This is startling since, of course, we have a fairly complete theory of elastic deformation as well as well-developed practical numerical methods of computation. But in Reuleaux’s time, the analogous theory of rigid motion was similarly well-developed; you can design a planar four-bar linkage to tween between three predetermined positions with compass and straightedge, and the whole lovely theory of quaternions commonly used for rigid motion in modern games engines predated this book by decades. As Reuleaux laments:
Here again is a point from which the weakness of the method hitherto employed can be surveyed at a glance. Its difference from the ideal method is not that it employs the inductive instead of the deductive method ; that would indeed be no advantage, but it might still be defensible. No, it has been entirely unmethodical. It has chosen no fixed method of investigation, or rather, it has not found any in spite of zealous search...
Rhetorically, this Introduction is a masterpiece; Reuleaux promises the world to his diligent student, while warning them of the difficulty of their quest and deprecating their existing knowledge:
We often do not know ourselves how closely wedged-in our ideas are by the boundaries which education and study have drawn around us. ... all these pile up mighty hindrances. I cannot therefore shorten the way, although the truths to which it leads are of great simplicity. ... to conclude in the words of Göthe, ” What is not understood is not possessed.”
It’s interesting to note that in Reuleaux’s time it was not yet known how recent the introduction of the wheel was, due to the still primitive state of the archaeological science; he claims “carriages” are known from the oldest times, while in fact they seem to postdate even such recent artifacts as the Great Pyramid.
His account here of Watt’s parallel motion is sticking with me. I keep thinking about it.
I just can’t get over the evocative nature of the prose, and the sharp contrast with the uniform oatmeal mediocrity of modern academic writing:
So soon as the force Q begins to act it calls forth in the interior of the wheel, the shaft and the supports, internal molecular forces, opposite in direction and exactly equal to it. ... there are opposed to all external forces others concealed in the interior of the bodies forming the system,...
after which he proceeds to quote a verse from Schiller (!).
Reuleaux actually comments a bit on the question of flexures here: “In actual machines...we use however only those materials which...alter their form under the action of external forces very little, so little that the corresponding variations from the original form may be neglected.”
I enjoy his description of the strength or rigidity of bodies as being “latent forces”, in contrast to actually existing “sensible forces”.
His definition of a machine is thought-provoking, as much for the aspects it omits (material handling, information, control, energy, force, thermodynamics, friction, efficiency, metrology, electricity, programmability, strength, tolerances, troubleshooting, wear, reliability, cost, clamping) as for what it includes:
A machine is a combination of resistant bodies so arranged that by their means the mechanical forces of nature can be compelled to do work accompanied by certain determinate motions.
It’s a damned sight better than “a combination of simple machines”, though, and it doesn’t exclude the use of kinetic energy or hydraulics as many inferior definitions have.
It’s interesting that this more or less explicitly excludes the boilers and condensers of the steam-engines Reuleaux has given such prominent placement earlier.
Here we see again the primacy Reuleaux grants to position and motion, but also what a high priority he places on forces. So far there’s no mention of compliance and backlash except as an enemy; nothing about springs or preloading.
Even with the limited definition Reuleaux has given, this section heading seems amazingly ambitious.
The moving bodies are prevented, by bodies in contact with them [emphasis in original]... this contact ... must take place continually, ...
This seems to exclude backlash and clearances entirely!
I have no idea what a “plummer block” is.
This is the section where he introduces the kinematic pair.
It’s fascinating that in is Figure 4, the first kinematic pair he introduces (before he even introduces the term) is not any of the basic ones, but rather a block sliding in an irregularly curved plane channel as the channel constrains it to rotate unevenly. Then on the next page his figures 5 and 6 are classic screw and prismatic joint.
Aha, and after Figure 9 he explains the meaning of “higher kinematic pair”:
Accordingly the reciprocal combinations of two elements gives us again a pair of elements, which may differ from either of the single pairs of which it is composed.
(And, as with his Figure 4 example of the weird sliding block, his first explicit example of a higher kinematic pair is an arbitrary weird thing, rather than something well-known.)
I really appreciate that Reuleaux is giving one or two examples first and only then giving definitions.
Also, he defines kinematic chain here, far more clearly than I’d heard it defined before. So I’m finding this reading very rewarding so far.
I’d never thought of this before, but there’s an interesting analogy between how changing the position of one element in a closed kinematic chain (a term I’m not sure I understand completely yet) alters the position of all the others, and how changing the current or voltage of one element such as a capacitor or inductor in an electrical circuit alters the current and voltage of all the others. In both cases you can’t analyze a single component in isolation; to see the system non-wholistically you must find other elements into which to decompose it, such as eigenstates or linearly superimposed circuits. But I don’t know what those components could be for either a closed kinematic chain or a nonlinear electrical circuit. (And I don’t think anything similar to the linear decomposition of circuits is known for kinematic chains, or things like the linkage-synthesis papers I’ve been reading would be using it.)
The aside “a cylindrical pin fitting a corresponding eye, the axes of all being parallel” immediately calls to mind Hoberman spheres and the like; if the axes instead all intersect at a point, you get the same sort of linkage, but thus constrained to a sphere rather than a plane. And of course there are other relationships that can provide more interesting movements. (Hoberman’s insight was similar but had to do with lines drawn transversely through the centers of multiple such joints.)
Aha, and here we have another key definition of Reuleauxian kinematics:
A closed kinematic chain, of which one link is thus made stationary, is called a mechanism.
At this point it’s worth comparing 19th-century European automata with 19th-century Japanese automata; while both treat the outward form and appearance of the mechanism as being as important as its position, work, and movement, the Japanese automata extensively used cable drive, including on uneven cams, while the European automata like Jaquet-Droz’s Writer used exclusively rigid bodies. Reuleaux is seeking to embrace not only pulleys and belts in his analysis but even hydraulic machines, but he has barely mentioned anything flexible or hydraulic yet, despite the world-shaking importance of the mechanization of textile manufacture in the late 18th and early 19th century, in part by Vaucanson himself.
So we can see in some sense why Babbage’s work was so unsuccessful and why the Writer would not be equaled for some 150 years, roughly until Bush’s Differential Analyzer. To keep the world of machinery intellectually manageable, its paradigm deliberately excluded consideration of the aspects crucial to such achievements. In Reuleaux so far I see not even the smallest hint of the approach that animated Zuse, Shannon, and Turing.
Here we see boldface in the text for a definition:
The effort thus applied performs mechanical work which is accompanied by determinate motions; the whole, that is to say, is a Machine.
We also see the first mention of clocks, though nothing about the questions of metrology, calibration, and cancelation of errors that had enabled the conquest of Latitude with machinery a century before, as well as the first mention of balances (in the sense of a weighing-scale); though he does promise to “consider these questions systematically” later.
One of the words whose usage has apparently changed significantly since this book was translated is “empirically”, which seems here to mean “by trial and error”.
He does finally mention “the spinning-machine” in this section, and the sewing-machine. The thread and fabric of the sewing-machine seems to fit very poorly into Reuleaux’s theory — the whole machine exists to put them into certain regular motions with respect to one another, but their motion is neither rigid nor determinate, and substantial parts of the machine exist purely in order to modulate the friction and tension on them.
“Phoronomic” is explained in the first section here; it means something like “concerning the study of the geometric motions of rigid bodies”.
Phoronomy, etc.
Newtonian relativity.
Relative motion of two points is considered only in terms of their distances, thus omitting rotation — rotational orientation is not considered proper to a point. Motion of a point relative to a plane it’s moving in requires only its distances from two points, not three, permitting a reflection ambiguity — perhaps chiral orientation is not considered proper to a plane.
Interestingly, it seems that when he considers motion of a line relative to another line, he does consider sliding motion along the line to be motion.
It’s surprising that he considers “the Phoronomics of point-systems” to be “exhausted” by propositions in two dimensions only!
This idea of a “temporary center” of rotation, for any arbitrary rotation, is very interesting. It reminds me of the compass-and-straightedge four-bar-linkage construction technique.
“Open polygons” occur frequently in computer graphics but they are usually called something else.
I don’t understand this “reciprocal polygon” yet. Why exactly must it exist, and be reciprocal?
Oh, this makes the reciprocal polygon thing a bit clearer, though it’s still not clear why it must exist. Boy, this sure is a different meaning of “centroid” than the one I’m familiar with, though not completely unconnected — the centroid of a uniform shape is its center of gravity, which is the center on which it turns when it has only angular momentum.
All this terminology of “common, curtate, and prolate trochoids” is new to me, and I think I’ll have to look it up.
When he says, “All relative motions of con-plane figures may be considered to be rolling motions”, he seems to be omitting the possibility of pure translational motion, which is the limit of rolling about an infinitely distant center.
It’s nice that he is getting into three-dimensional objects moving now, but I can’t help but wonder if there are possibilities of three-dimensional motion other than the possibilities arising from extending two-dimensional motions prismatically, although Reuleaux claims there aren’t. For example, motions where the instantaneous axis of rotation twists from one moment to the next. And what is the instantaneous axis of rotation of a common screw?
It is now apparent that the reciprocal polygons of which Reuleaux was speaking may have quite different shapes.
The construction he gives for finding the second point M₁ in Figure 19 (p. 66) escapes me at the moment. I will have to come back to this.
Oh. Because M₁ is notionally part of the same rigid body as P and Q, its distances to them are not changed by rolling; at the given point in the movement has been brough into coincidence with O*₁, so those distances are the distances from the moved positions P₁ and Q*₁.
Figure 20 in the Google scan (p. 67, 88/651) is partly illegible; it is identical to Figure 10. In the Archive’s scan it is perfectly legible. Incidentally, I think this page shows that the scans were taken from two separate physical copies, as the Archive’s scan is stamped, “Reese Library of the University of California”.
Aha, and here he confronts the question of pure translational motion, using the device of “infinitely distant points”, which seems pretty kosher, although not all this geometry so far is projectively invariant. (Or is it?)
Figure 22 is blowing my tiny fucking mind.
I am totally not understanding this. I thought the motion uniquely determined the two centroid curves? But now we can invent more centroids for the same motion?
Here he confronts the question of parallel motion in all of its complexity.
Hmm, there’s a clue about these “secondary centroids” in the part where he talks about gear teeth (though he never says “gear”, always “spur-wheel”).
Now Reuleaux seems to be taking up the question that had worried me earlier, that of motions in space that are not merely the extrusion of motions in a plane.
Whoa, trippy, dude.
Okay, great, now we’re getting into screwing motions?
Oh wow.
Holy shit.
Hmm, he has a hypoid drive here, although he doesn’t call it that.