The Kinematics of Machinery/Introduction

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The aim of the following chapters is to determine the conditions which are common to all machines, in order to decide what it is, among its great variety of forms, that essentially constitutes a machine;—they are therefore called Outlines of a Theory of Machines. The whole study of the constitution of machines—the Kinematics of Machinery—naturally divides itself into two parts, the one comprehending the theoretical, and the other the applied or practical side of the subject; of these the former alone forms the subject of this work. It deals chiefly with the establishment of those ideas which form the foundation of the applied part of the science, and in its treatment of these its method differs in great part essentially from those hitherto employed.

As I have here to do chiefly with theoretical questions, it might seem that I could hardly expect to interest others than those concerned only with the theoretical side of this special study. But Theory and Practice are not antagonistic, as is so often tacitly assumed. Theory is not necessarily unpractical, nor Practice unscientific, although both of these things may occur. Indeed in any department thoroughly elucidated by Science the truly practical coincides with the theoretical, if the theory be right. The popular antithesis should rather be between Theory and Empiricism. This will always remain, and the more Theory is extended the greater will be the drawback of the empirical, as compared with the theoretical methods. The latter can never be indifferent, therefore, to any who are able to use them, even if their work be entirely "practical," and although they may be able for a while longer to get on without them. The theoretical questions, however, which are here to be treated, are of so deep-reaching a nature that I entertain the hope that those who are practically, as well as those who are theoretically concerned with the subject, may obtain help from the new method of treating them. I am thus obliged to lay before both equally the grounds upon which I have given up the customary ideas upon the subject and put others in their place.

In attempting to place the theory of the constitution of the machine upon a new basis I do so with the conviction that my trouble will be repaid only if it prove of some actual advantage in the right understanding of the machine. I venture to promise such a result with confidence. He who best understands the machine, who is best acquainted with its essential nature, will be able to accomplish the most by its means. It is not a matter of merely setting forth in a new form and order what is already well known, or of substituting a new classification and nomenclature for the old. Possibly with such improvements the subject might be more conveniently and elegantly taught, but for practical purposes the old forms might be used for a long time to come. On the contrary, if the new theory is to lay claim to general interest, it must be capable of producing something new; it must make problems solvable which before could not be solved in any systematic way. This may certainly be said to be the case if it succeeds in making Machine-Kinematics, down to its simplest problems, truly scientific.

This subject has indeed, in a certain sense, been scientifically treated hitherto, in so far, namely, as particular portions of it admit of mathematical treatment. But this concerns a part only of the subject, and not that part which is peculiarly its own; so far as the treatment has been scientific, too, it has been mathematical or mechanical, and not kinematic. This last in its essence, in the ideas belonging specially to it, has been left indistinct, or made clear accidentally at a few single points only. It is like a tree which has grown up in a dark tower, and thrown out its branches wherever it could find an outlet; these, being able to enjoy the air and light, are green and blooming, but the parent stem can only show a few stunted twigs and isolated leaf-buds.

The mathematical investigations referred to bring the whole apparatus of a great science to the examination of the properties of a given mechanism, and have accumulated in this direction rich material, of enduring and increasing value. What is left unexamined is however the other, immensely deeper part of the problem, the question: How did the mechanism, or the elements of which it is composed, originate? What laws govern its building up? Is it indeed formed according to any laws whatever? Or have we simply to accept as data what invention gives us, the analysis of what is thus obtained being the only scientific problem left—as in the case of natural history?

It may be said that the last method has been hitherto followed exclusively, for only traces appear now and then of penetration behind these data. The peculiar condition consequently presents itself throughout the whole region of investigation into the nature of the machine that the most perfect means have been employed to work upon the results of human invention—that is of human thought without anything being known of the processes of thought which have furnished these results. Terms have been made with this inconsistency, which would not readily be submitted to in any other of the exact sciences, by considering Invention either avowedly or tacitly as a kind of revelation, as the consequence of some higher inspiration. It forms the foundation of the kind of special respect with which any man has been regarded of whom it could be said that he had invented this or that machine. To become acquainted with the thing invented we leap over the train of thought in which it originated, and plunge at once, designedly, in medias res.

If, for instance, we consider the well-known parallel motion which Watt invented for his steam-engine, or that of Evans, or that of Reichenbach, according to the method hitherto adopted, we find that after we have classified them we have nothing further to do than to ascertain the laws of motion by which they are governed as mechanisms, to fix upon the constructive forms most suitable for them—and, if it be required to go further, to elucidate their more intimate mutual relationships. How, however, their inventors arrived at them we leave unentered upon, except in so far as our personal feeling of interest in this point is concerned. Now and then we are glad to overhear the Genius in his thought-workshop, but more from curiosity than in any spirit of investigation. And yet it would appear from what we have said that here there is a great step further to be made. Let us try.

Watt has left behind for us in a letter some indications of the line of thought which led him directly to the mechanism just alluded to. "The idea," he writes to his son in November 1808, "originated in this manner. On finding double chains, or racks and sectors, very inconvenient for communicating the motion of the piston-rod to the angular motion of the working beam, I set to work to try if I could not contrive some means of performing the same from motions turning upon centres, and after some time it occurred to me that A B and C D being two equal radii revolving on the centres B and C, and connected together by a rod A D, in moving through arches of certain lengths, the variation from the straight line would be nearly equal and opposite, and that the point E would describe a line nearly straight, and that if for convenience the radius C D was only half of A B, by moving the point E nearer to D the same would take place, and from this the construction, afterwards called the parallel motion, was derived. .... Though I am not over anxious after fame, yet I am more proud of the parallel motion than of any other invention I have ever made."

The Kinematics of Machinery Fig 1.png

Fig. 1.[1]

Interesting as this letter is, a closer examination of it reveals a deficiency which perhaps the questioner also may have discovered. We quite appreciate the motives as well as some of the final results of Watt's exertions, but we obtain no indication of a methodical train of ideas leading up to them. Moreover it must be remembered that the description is written twenty-four years after the invention, that therefore reflection and recollection have had time to work upon each other. Watt expressed himself much more directly and distinctly in a letter written in 1784 to Boulton, giving him the first idea of the invention:—

"I have started a new hare," he writes; "I have got a glimpse of a method of causing a piston-rod to move up and down perpendicularly by only fixing it to a piece of iron upon the beam, without chains, or perpendicular guides, or untowardly frictions, arch-heads, or other pieces of clumsiness, by which contrivance, if it answers fully to expectation, about 5 feet in the height of the [engine] house may be saved in 8 feet strokes, which I look upon as a capital saving; and it will answer for double engines as well as for single ones. I have only tried it in a slight model yet, so cannot build upon it, though I think it a very probable thing to succeed, and one of the most ingenious simple pieces of mechanism I have contrived, but I beg nothing may be said on it till I specify."[2] 1

If we examine the specification referred to we find no less than six methods of guiding described, and among them the very "perpendicular guides" and "arch-heads" found fault with above; two of these methods take the form which our mechanism can assume. One of the six is specially notable,—it is exactly the parallel motion of Reichenbach, and seems to lead up to the motion more generally known by Watt's name. Watt has evidently not recognised this,—and later on it has completely escaped him, at which one cannot wonder, considering the uncouth garb of timber beams and hammered rods in which the elegant mechanism was at that time disguised.

We see that even a thinker like Watt was at fault in the essential elucidation of the matter. Yet we note at the same time that each thought in the inventor's stream of ideas is developed out of another, that the ideas form a ladder up which he presses step by step,—through labour and exertion,—to his goal. His eventual success gains from us the more respect that he did not find the end of his exertions close at hand. But the inspiration, the instantaneous illumination,—cannot be detected;—he says above "and after some time it occurred to me," which only points to previous uninterrupted search, continuous following-out of thoughts. "By continuous thinking about it," answered Newton to the question how he had discovered the law of gravitation. Göthe also gives us the same idea in his sentence,

"What is Invention? It is the end of seeking."

The links which connect isolated thoughts seem indeed to be almost entirely destroyed,—we have to reconstruct them. We see the whole before us only like a faintly outlined or half-washed-out picture, and the painter himself can hardly furnish us with any better explanation of it than we can discover for ourselves. Indeed the comparison holds good in more than one point. In each new region of intellectual creation the inventor works as does the artist. His genius steps lightly over the airy masonry of reasoning which it has thrown across to the new standpoint. It is useless to demand from either artist or inventor an account of his steps.

Observations similar to those made in this single case can be made throughout the whole history of invention, wherever the genius of past generations has busied itself in bringing forward new things. The invention of the steam-engine, for example, stretches back through a whole century,[3] without ever making a step in advance of the natural development going on in other departments of knowledge.

In the school of Galileo,—where his experiments on falling bodies first threw a ray of light through the scholastic cloud which had veiled all knowledge,—there began early in the seventeenth century those experiments in physical science with the growth of which the invention of the steam-engine is inseparably bound up. It is no mere chance that the place is distinguished also as the centre of great artistic development;—art and science nourish together on a rich soil. It is as if the proud citizens of Pisa had made their marble tower to lean expressly for Galileo's experiments. In Florence (1643) Galileo's disciple Torricelli, still in the freshness of youth, made his discovery of the heaviness of the air,—on which there followed directly alarming outcries for the preservation of the "horror vacui" and the whole threatened belongings of the wiseacres of the time. The centre of disputation and investigation passed from Tuscany to France when Pascal took up the question in 1646,—he, after a while, came quite over to the new ideas. He caused the memorable first barometric measurement to be made on the Puy-de-Dôme in 1648. It was conclusive, and the joyful bells of Münster and Osnabruck rang in the triumph of the young science.

The line in which the centre of this newly-awakened activity moved now turned to the north-east, towards Germany, into the country of the great Electors. Tilly had not been able to destroy the intellectual life of Magdeburg. There in 1650 Otto von Guerike brought a new idea into what was the question of the day in physical science,—that namely of the possibility of utilising the atmospheric pressure as a force. He showed this both popularly and scientifically with the air-pump and other experimental apparatus. The search for means of utilising the power stored in the atmosphere by some simple vacuum arrangement now began everywhere. For a long time no satisfactory results were obtained, but at last, in 1696, Papin at Marburg discovered the true solution—the condensation of steam in a cylinder fitted with a piston. The steam-engine was invented. Papin, an experimenter really worthy of respect, who attempted to solve the problem in the most various ways, and who has made a whole series of other remarkable inventions, is its true inventor. His arrangement, however, was as yet very incomplete and unpractical; the great ideas required still to be carried beyond learned circles and Latin treatises into practical life. In this Papin did not succeed; he never got beyond the beginning. His first large steam cylinder still stands incomplete as a monument in the court of the Museum at Cassel, but his ideas crossed the English Channel and found their way directly to practical men. The mechanics Newcomen and Cawley produced in 1705 a really useful pumping-engine, which soon found actual employment in mining operations.

The spirit of invention now rested for a while, as if exhausted with the exertions of the last few years. The pause was unavoidable, for the necessary means for making progress did not exist. It was necessary that more should first be known of heat, which could not yet even be measured. The thermometer had first to be perfected, the whole theory of heat had to be greatly advanced. Then came Watt (circa 1763), whose far-reaching genius furnished a whole series of mechanical and kinematic inventions, and brought the machine in a short time to a high degree of completeness, enriching at the same time all the kindred realms of knowledge. From that time onward the rapidly extending employment of the machine and its continual development, improvements from a thousand heads and a thousand hands, have brought it to its present perfection, and made it absolutely the common property of all.

I have entirely omitted in this hurried sketch what Humboldt called the "horrible wrangling about priority." From such a summary review we might almost be led to believe in the entirely spontaneous unfolding of ideas, were it not that each separate energetic step forward shows us the grasp of more distinguished endowments, and convinces us afresh of the importance of genius in the further growth of the race. Throughout the whole, however, we can discern the one idea developing itself out of the other, like the leaf from the bud or the fruit from the blossom; just as throughout nature everywhere each new creation is developed from those which have preceded it.

I believe I have shown in the preceding paragraphs that a more or less logical process of thought is included in every invention. The less visible this is from outside, the higher stands our admiration of the inventor,—who earns also the more recognition the less the aiding and connecting links of thought have been worked out ready to his hand. To-day, when scientific aids are so abundant in every branch of technical work, progressions of the greatest importance are frequently made without receiving any such high recognition as in former times. Everything lies so clearly and simply before us, that it can be reached and comprehended by commonplace intellects. At the same time the relative difference between the work of the commonplace and that of the highly cultivated intellect is even greater than before, and by this may be explained the apparently almost feverish progress made in the regions of technical work. It is not a consequence of any increased capacity for intellectual action in the race, but only of the perfecting and extending of the tools with which the intellect works. These have increased in number just like those in the modern mechanical workshop;—the men who work them remain the same.

Let us return now to our special subject, and examine in a more strictly historical way what has been done hitherto for theoretical Kinematics. The reader need not fear that I shall disturb the dust on old parchments in order to build up from dry dates the foundations of a science. We look now for the beginnings of the idea of our subject, and this delicate material can be taken from the ancient volumes without disturbing the moths.

In earlier times men considered every machine as a separate whole, consisting of parts peculiar to it; they missed entirely or saw but seldom the separate groups of parts which we call mechanisms. A mill was a mill, a stamp a stamp and nothing else, and thus we find the older books describing each machine separately from beginning to end. So for example Ramelli (1588), in speaking of various pumps driven by water-wheels, describes each afresh from the wheel, or even from the water driving it, to the delivery pipe of the pump. The concept "waterwheel" certainly seems tolerably familiar to him, such wheels were continually to be met with, only the idea "pump"—and therefore also the word for it—seems to be absolutely wanting.2 Thought upon any subject has made considerable progress when general identity is seen through the special variety;—this is the first point of divergence between popular and scientific modes of thinking. Leupold (1724) seems to be the first writer who separates single mechanisms from machines, but he examines these for their own sakes, and only accidentally in reference to their manifold applications. The idea was certainly not yet very much developed. This is explained by the fact that so far machinery had not been formed into a separate subject of study, but was included, generally, under physics in its wider sense. So soon, however, as the first Polytechnic School was founded, in Paris in 1794, we see the separation between the study of mechanisms and the general study of machinery, for which the way had thus been prepared, systematically carried out.

The completion of this separation connects itself with the honoured names of Monge and Carnot. The new line of study appeared first as a subdivision of descriptive geometry, from which it has only by degrees released itself. It fell to Hâchette to give instruction in it, and he, working upon outlines given by Monge, constructed (1806) a programme of which Lanz and Bétancourt afterwards filled up the details in their "Essai sur la composition des machines" (1808). Monge had entitled the subject "Elements of Machines," which he intended to be equivalent to means for altering the direction of motion. He understood by these "means" mechanisms, and based on this idea the arrangement of mechanisms according to the possible combinations of four principal kinds of motion, viz.: continuous and reciprocating rectilinear, and continuous and reciprocating circular. Omitting repetitions, these give ten classes of mechanisms, corresponding to the changes of motion between

Continuous rectilinear and continuous rectilinear
reciprocating ' '
continuous circular
reciprocating ' '
Continuous circular and reciprocating rectilinear
continuous circular
reciprocating ' '
Reciprocating rectilinear and reciprocating rectilinear
' ' circular
Reciprocating circular and ' ' ' '

This scheme,—or "system,"—was capable of extension,—and in the second edition (1819) it was extended by the addition of other fundamental motions,—continuous and reciprocating motions in curved lines,—increasing the number of classes from 10 to 21, but not altering the principle of classification. This indeed has remained with unimportant alterations in tolerably general use until the present time, and has thus acquired the sanction of very general recognition. Hâchette himself, who assisted in the production of Lanz' work,[4] adopted it unconditionally in his Traité élémentaire des machines, published in 1811. Borgnis, in his Traité complet de mécanique (1818) departed from it to a certain extent; he considered the problem more generally than his predecessors, and divided the parts of machines into six classes, which he called récepteurs, communicateurs, modificateurs, supports, régulateurs, and operateurs respectively. He did not take the change of motion as a leading principle, but used it only in determining the subdivisions. His system has not been accepted as necessarily antagonistic to Monge's, his method of division being taken as more or less suitable rather for the general study of machinery than for Machine-Kinematics. One leading idea at least of Borgnis' scheme has since become universally familiar;—his division of machinery into the parts receiving effort, the parts transmitting it, and the working parts. Through the brilliant works of Coriolis3 and Poncelet4 these have become supporting pillars, one might almost say articles of belief, in the modern study of machines. At the risk of being considered a heretic, I must say here that these fundamental notions require essential modification. The honoured Nestor of applied mechanics must pardon me for the sceptical saying: Amicus Plato, sed magis amica veritas.[5] We shall further on obtain the means of putting Borgnis' ideas to the proof,—but it is clear that his principles play too important a part in reference to the motions of the different parts of machinery to be altogether foreign to the study of Mechanisms. Borgnis' work itself is to-day quite out of reckoning,—his classification of machines and their parts has borne little fruit;—it serves for the most part as little more than a somewhat systematic exercise of the reader's memory. Nevertheless we shall find later on that more lies behind some of his thoughts than has been commonly supposed.

The year 1830 saw a notable change in the position of the study of Mechanisms, through the critical examination of its principles by the great physicist Ampère in his Essai sur la Philosophie des Sciences. In his system of sciences Ampère ranked the study created by Monge and Carnot as one of the third order, and attempted to lay down its exact limits. He considered in connection with it Lanz's treatise, and said, among other things—"It (this science) must therefore not define a machine, as has usually been done, as an instrument by the help of which the direction and intensity of a given force can be altered, but as an instrument by the help of which the direction and velocity of a given motion can be altered." He completely excludes forces from the investigations proper to the science, and says further, "To this science, in which motions are considered by themselves as observed in the bodies surrounding us, and specially in those systems of apparatus which we call machines, I have given the name Kinematics (Cinématique), from κινημα, motion." He further on encouraged the treatment of the science in text-books, for which he foresaw great use,—but he did not enter into any further details regarding it.

The seed thus sown by Ampère has borne rich fruit,—the Science of Kinematics was soon taken up (in France first of all) as a separate study, and a literature for it came rapidly into existence. The proposed name met with the most ready acceptance in France, and has since become more or less familiar in many other places.5 In the scientific limitation of the nature and aims of the study, however, the clearness so much to be desired has by no means been attained.

The next important original work is the "Principles of Mechanism" of the late Professor Willis (1841), a remarkable book, full of valuable illustrations from applied Kinematics, and of thoughts in relation to their real connection with each other. In system Willis differs from Monge. He considers that the scheme of Lanz, "notwithstanding its apparent simplicity," must be considered "a merely popular arrangement." He finds further in Lanz and Bétancourt a contradiction of Ampère's definition, in their inclusion of waterwheels, windmills and so on, among mechanisms, and will only allow that those mechanisms are pure which consist entirely of rigid bodies. With these mechanisms he lays special stress upon the important characteristic that they do not determine the actual motions in direction and velocity, as Monge says, but only the relations in direction and velocity between the motions occurring in the machine. According to whether these relations in any mechanism are constant or varying, he placed it in one or other of three classes, each of which has subdivisions corresponding to the means used for transmitting the motion (rolling-contact,—sliding contact, &c.)

Willis' observations bear throughout the mark of careful and earnest investigation, but while there is much in them that is true, there are also some things that are incorrect, as especially the exclusion of hydraulic and other such machines. To this I will return further on. It is worthy of note, however, that Willis' classification has never taken root in his own country, but that much more commonly people have been content to follow the well-trodden path marked out by Lanz.[6]

In Italy, Ampère's seed has also taken root. In the Cinematica applicata alle arti, a text-book for technical schools, which first appeared in 1847 under a somewhat different title, Giulio has left to his country a valuable gift. This book unites admirably Kinematics and Mechanics; it follows Willis pretty faithfully in essentials, but not without an attempt,—incompletely successful,—to replace the hydraulic machines which Willis had struck out. A delicate intellectual inspiration breathes through the whole work, and is the more notable that it was written for pupils not having more than an elementary acquaintance with mathematics. The concise mathematical expressions, which speak for themselves, have thus had to be replaced by explanations in words,—a method which presupposes a deeper understanding on the part of the author than is shown in many books bristling with formulæ.

In 1849, Laboulaye, again in accordance with Ampère's suggestion, attempted in his Cinématique to set forth the science of Mechanisms in a complete form. He also discards Willis' limitation of mechanisms to those constructed of rigid bodies, and points out also that Ampère required something impossible in wishing absolutely to exclude the consideration of force from Kinematics. Besides this he attempts to determine a new theoretical method of a general character. This consists in dividing the whole "machine-elements" into three classes, which he calls système levier, système tour, and système plan, corresponding respectively to the making fixed (inébranlables) for the time, one, two, or three and more points of the moving body. These "systems," however, do not really cover the problem, of which we shall find the proof in its proper place. Even their originator has not made any real use of them, feeling, doubtless, that there was not much to be gained by doing so. So far as applied Kinematics is concerned, he seems rather to return to the system of Lanz, with suitable subdivisions. Indeed, he goes so far in this direction as to construct Monge's system a priori, thus showing that it forms, in reality, the groundwork of his theory. Laboulaye has done no service to the science of Kinematics by this philosophical experiment, for by the apparently convincing form of his proof he has prevented those familiar with the subject from making further investigations. His a priori construction is based entirely upon the motion, of a point, and is for that very reason inapplicable to the motions of a body, or system of points. In other matters, Laboulaye's book is valuable, and it has without question been the means of widely spreading much useful information; it relies in the practical part confessedly very much upon the infinitely industrious Willis, with whom it sometimes even shares errors.

Morin also, in a little book (1851) intended for elementary instruction, has made a collection of the principles of Kinematics, called, in the later editions, Notions géométriques sur les mouvements. It is unpretending, and written in a very intelligent manner, and contains some capital leading thoughts, but in essentials it follows Monge's scheme.

In Germany it may almost be said that nothing was done during the period under consideration for the development of theoretical Kinematics. Weisbach, in his article "Abänderung der Bewegung," (Alteration of Motion), in Hülsse's Encyclopaedia (1841), adhered altogether to the system of Lanz,—his own scientific work however had admittedly quite a different direction. Something new might have been expected from Redtenbacher, whose work was so continually connected with mechanisms. His highly philosophic brain perceived strongly the insufficiency of Monge's system; but drawn away from the subject, first in the development of scientific machine construction, and afterwards by his work in mechanical physics, he abandoned it, but did not bring anything new into its place. This was probably his reason for holding that no true system of the study of mechanisms was possible, that they could be arranged only according to their practical usefulness, and for the rest must be treated mathematically. This nihilism may be read between the lines of his valuable work, Die Bewegungsmechanismen (1857), in which he describes and treats theoretically the mechanisms of the collection of models at Karlsruhe. That this book, systemless as it is, has had no inconsiderable circulation, shows that our technical public feel the lively necessity which exists for the theoretical exposition of the subject.

In France, meanwhile, progress which has been of great importance to Kinematics has taken place in the region of Geometry. The geometrical method of treating the motion of rigid bodies developed by Euler in the last century was again taken up, and soon further extended, by Chasles and (especially) by Poinsot. The works of the latter,—Théorie de la rotation des corps, and Théorie des cônes circulaires roulants, gave a great impulse also to the employment of geometrical methods of representation in the study of mechanisms. The propositions of Euler, which had hitherto possessed no more than a purely theoretical and abstract interest, were now formed by the French kinematists into fundamental doctrines. They breathed fresh life into what had become a somewhat dull study. Under their influence appeared Girault's Elémens de géom. appl. à la transformation du mouvement 1858,—Belanger's Cinématique 1864,—and Haton's Traité des mécanismes 1864; the two first specially rich in geometrical, i.e. in theoretical parts, the third bearing rather on the application of theory to the mechanisms themselves. All three books, however, valuable and important though they be, fall into the old difficulties as to classification as soon as they enter upon the applications of the science. They all differ from Monge, for the inadequacy of the "ancien système" became too evident to escape detection in the light of the new ideas,—nevertheless they all remained, for better or worse, partly involved in it. They differ too among themselves, and each seems to hesitate in his own particular way between Monge and Willis. Girault and Belanger take their principal divisions from the changes of motion,—but in entirely different ways,—using the various methods of transmitting the motion as subdivisions, Belanger with the addition of Willis' relative velocities. Haton recognises the want of the old system, and points out, for example, that to arrange toothed wheels according to it would require in all some 21 different classes;—he himself takes his principal divisions from the methods of transmission; these he divides into nine classes, of which Rollers, Guide-bars, Eccentrics, Toothed-wheels, Connecting Rods, and Cords form the first six, the three last bearing in common the fatal designation appareils accessoires. One whole third of the subject therefore has been cut off,—placed as it were like a note below the text.

Nor has this new geometrical development, into the further growth of which I need not enter, been able to bring about any common method of treatment for applied Kinematics. Something still remains doubtful, and the results obtained have correspondingly been uncertain. Even those who wish to employ pure scientific methods only seem to be convinced that such methods cannot be used in the applied part of the science. They fall into Redtenbacher's nihilism, and cut "Cinématique pure" away from "Cinématique appliquée." Résals' Cinématique pure (1861) is an example of this,—it shows that the evaporation, as it were, of kinematic problems into those of pure mechanics can scarcely, with such a method of treatment, be avoided.

Moreover, as a further fruit of this uncertainty there has been an attempt to construct 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 subdivide his subject into pure and applied parts, as fully described in his Essai sur l'automatique pure, 1863. Stamm has earned considerable distinction in connection with the self-acting spinning-machine,—that is to say in a special case of applied Kinematics,—which those technically concerned know how to appreciate. His separation of Automatics from the science to which it belongs must, however, be considered unpractical; it cannot exist by itself,—it is only a portion of a synthetic method based upon the fundamental laws of Kinematics, to which science it therefore belongs inseparably.

We have reached the end of our literary review.[7] We have found, on the one hand, a most unsatisfactory confusion of attempts to find a form for one and the same circle of ideas. As many systems as authors, no resting point reached, always new trying and seeking,—and as a conclusion we have Ampère's well-defined science split into two, indeed into four sciences, as if it were one of those Infusoria which are propagated by division. On the other hand, we may make the consolatory remark that the observations have grown more and more in exactness and delicacy, as also that both the methods of examination and the mechanisms examined have by degrees greatly increased in number. The two sides of the question, the theoretical and the applied, have thus carried on a separate existence side by side; but their union has not yet been accomplished. The cause of this must be sought in the systems alone, for the applications, the mechanisms themselves, have been quietly developed in practical machine-design, by invention and improvement, regardless of whether or not they were accorded any direct and proper theoretical recognition. Indeed the theories used hitherto have contributed to this development only in regard to the form of execution in certain cases, as e.g. in the methods of drawing wheel-teeth, &c.; they have however furnished no new mechanisms. This circumstance is very remarkable; it explains the conservative tenacity with which practical mechanists, where they have adopted any theory whatever, have always fallen back upon the old and apparently natural ideas of Monge in spite of the promises held out by the more recent theories; this the technical journals everywhere sufficiently show.

I believe I have proved the insufficiency of the theoretical Kinematics hitherto taught, and the necessity for some reform. The question now comes, wherein exactly does the error of the existing methods lie?

Monge's classification, however natural it appears, does not in the first place correspond to the real nature of the matter. Did it really do so—did it resemble, for instance, the classifications of Linnæus and Cuvier in organic nature—it would, like them, be able to make its footing firm; and the obstinate conservatism above referred to may perhaps be explained by the existence of a dim feeling that some analogous relation must also exist here. But this is not the case. Even if it were supposed that the function of the science did not extend beyond arranging the mechanisms in classes, still their division cannot be made according to the changes of motion, for this would necessitate endless repetitions. Almost all mechanisms might be looked for, and might also be placed, in at least two such classes, and most in from four to six or even ten to fifteen, for they can be used, and in practice are actually employed, for just so many different kinds of change. Willis, who strictly investigated this very point, and had a strong perception of the necessity for logical order, showed a distrust of the elasticity of the basis of his own system, so that his classification does not carry conviction with it. He wishes to adhere consistently to his fundamental principle of relative motions, and so finds it necessary to treat together very various kinds of mechanisms; and as a single mechanism frequently contains in itself several kinds of relative motions, he is compelled repeatedly to enter upon repetitions of an extended kind. Other objections might be raised against the modified classifications of Laboulaye, Girault, Belanger, Haton and the others, generally as well as individually, for no true science can be moulded at will in six or eight different ways.

The real cause of the insufficiency of the system is not, however, the classification itself; it must be looked for deeper. It lies, as I have already pointed out, in the circumstance that the investigations have never been carried back far enough,—back to the rise of the ideas; that classification has been attempted without any real comprehension being obtained of the objects to be classified. The formation of a science cannot be entered upon in medias res; it requires that we should start, as in mathematics, from the very simplest elements, the axiomatic beginnings. Without the determination of these the goal can never be reached. A single trial of the method commonly employed shows this very clearly.

In the old classification a commencement was made very commonly with the changing of one rectilinear motion into another; but no one asked whence the first rectilinear motion came, why it existed, how it had been created. To take a special case, Hâchette and Lanz choose for their first mechanism the so-called "fixed pulley." In this case it is the rectilinear motion of the cord as it runs off the pulley which is changed into another such motion in the part of the cord running on in the opposite direction. Why, however, the first motion is rectilinear we do not understand. Indeed it is not necessarily rectilinear, for the cord running off may be pulled to one or the other side, if it only be kept stretched, without in any way altering the mechanism. Then also the motion of the cord originates in the circular motion of the points of the pulley or drum; only after this motion do we have that of the cord itself. Thus the very first problem takes us beyond the limits of the class. The same indeterminateness which has been pointed out in the motion of the one cord belongs also to the other. We see therefore that, even in the very first example, inexactness in a number of ways makes its appearance. It may be noticed too that this first problem of the fixed pulley, so far as concerns the theory of the motions involved in it, is already a very complicated one, as I shall show in the text later on.

The ideas by which alone the nature of a simple mechanism can be arrived at may be very complex, or in certain circumstances may be quite the reverse. But equally whether these ideas be simple or complex, if it be desired to examine and understand the simple apparatus scientifically, it is necessary to work through the whole succession of them, passing from each one to the next higher from which it was developed, until really general principles are arrived at. However difficult this may be, and however little use it may appear to have, it must be done; its omission by kinematists hitherto has caused the wreck of all their theories. What they have done is what I have already indicated as incorrect,—they have accepted the simple or apparently simple mechanism as it came from the hand of the inventor, whether he were a well-known person, or nameless in the traditions of the dawn of national history.

An examination of these hazy traditions gives the kinematist much subject for remark. Denying myself here the prosecution of this attractive subject, upon which I shall enter in some detail further on, I must dismiss it with one remark. Following back machines to the earliest forms in which they have been used in historic times, we find in different places contrivances of various kinds in use,—from somewhat complicated machines to the very simplest arrangements,—all of which must be called "machine." We are not yet in a position to discuss the criterion of the comparative difficulty of their invention; we require here only to note that they appear in various places independently of each other. The rollers upon which the Assyrian as well as the Egyptian builders moved their enormous stones are among these primitive machines; carriages of wood and metal, both for war and for transport, were possessed by the Egyptians, Babylonians and Indians in the remotest antiquity; water-wheels were in use in old Mesopotamia and in Egypt, as well as in China, India and Central Asia; toothed wheels were known to the Greeks, as were also the screw, the pulley, certain systems of levers, &c. Some of these arrangements have come down to us unaltered; others, however in the way already enlarged upon, formed only steps leading up to their present successors. All have been thought out by human brains,—now and then by brains of special capacity, and then praised as God-sent gifts,—but in all cases thought out, produced by a mental process which has contained more or less well-defined gradations.

And to-day just as formerly they must still be arrived at by a mental process; and this forms the problem which it must be the chief aim of theoretical Kinematics to solve. So long as it could not reach the elements and mechanisms of machines without the aid of invention, present or past, it could not pretend to the character of a science, it was strictly speaking mere empiricism—(sometimes even of a very primitive kind),—appearing in garments borrowed from other sciences. When however its investigations enable it to furnish the means of producing any required kind of motion, it will begin to deserve the name of Science. It will then itself point out the true classification of its own material. It can put before itself the question as to the change of one motion into another, and decide as its own director whether really and to what extent a division founded upon this is important. As a genuine science, moreover, it will find its laws in itself, and require no Lycurgus to deliver them from without.

Here we reach another weighty and notable consequence. If the processes of thought by which the existing mechanisms have been built up are known, it must be possible to continue the use of these processes for the same purpose;—they must furnish the means for arriving at new mechanisms—must, that is to say, take up the position hitherto assigned to invention. I hope not to fall under the suspicion of saying anything so absurd as that the new method will make it possible for the commonplace head to become oracularly inventive like that of the genius. The case is rather that it will become possible to introduce into machine-problems those intellectual operations with which science everywhere else pursues her investigations. I have attempted already to show that Invention, in those cases especially where it succeeds, is Thought; if we then have the means of systematizing the latter, so far as our subject goes, we shall have prepared the way for the former.

Göthe,—who had so great an interest in the inner nature of everything which could enlarge the circle of our ideas,—expresses himself in the following noteworthy sentence: "Everything that we call Invention, discovery in the higher sense, is the ultimate outcome of the original perception of some truth, which, long perfected in quiet, leads at length suddenly and unexpectedly to productive recognition." Schopenhauer too, whose thoughts not unfrequently seem to take the same direction as those of Göthe, says upon a very similar question:—"Our best, most able and deepest thoughts often seem to enter our consciousness like an inspiration,—sometimes directly in the form of a weighty sentence. Evidently, however, they are the result of long and unconscious meditation, and numberless long past and often entirely forgotten thoughts and conclusions. The whole process of thought and conclusion seldom lies on the surface,—seldom takes, that is, the form of a chain of reasoning clearly thought out, although we may endeavour to attain this in order to be able to give ourselves and others an account of what has occurred: commonly, however, the rumination by which the material received from without is converted into thoughts, occurs as it were in darkness,—taking place almost as unconsciously as the change of the nourishment into the fluid and substance of the body. Thus it comes about that we can often give no account of the origination of our deepest thoughts;—they are born from our most secret being. Out of its depths arise unexpectedly Opinions. Ideas, arid Conclusions."6

There is nothing, however, impossible in the ideas necessary for the origination of a mechanism being "clearly thought out," and they can then lead to what is sought just as in mathematics the clearly-reasoned and well-connected ideas lead up to the result. In other words, the invention of a mechanism will be to the scientific kinematist a synthetic problem,—which he can solve by the use of systematic, if also difficult, methods. The clever man, supplied with such powerful instruments, will leave the less clever behind in future as hitherto,—just as the mathematical genius leaves behind the mere algebraist who works only with operations learnt by rote.

The thorough understanding of old mechanisms, however, is even more important than the creation of new ones. It is indeed astonishing to how small a depth the methods hitherto used have penetrated into their real nature, and how incompletely known therefore are most of the mechanisms in common use. To the thorough, thoughtful mechanician who looks at his work as a serious matter, a scientific investigation of the Kinematics of Machinery will in this respect be specially valuable. It will relieve him of the minute and often worrying search after solutions of his problems by rendering it possible for him to work systematically. The technologist too, who hitherto has scarcely made any use of Kinematics, will find in it an important assistance in understanding old machines and devising new ones. The deepening of the comprehension which must occur in such cases as these renders it certain that the remodelled science will take its share in the real end before us,—the progressive development of the machine.

If we look back over the representation I have tried to give of the way in which the subject has hitherto been treated, and of the Ideal to be aimed at, the old method appears to have no inner unity, although the scientific methods of investigation which it employs have prevented this from being generally recognised. We have, however, shown that these are only of secondary importance compared with the establishment of the special ideas and principles peculiar to the subject. This question, moreover, must at once be seen to be one which actually concerns a department of investigation belonging distinctly to the exact sciences. This being recognised, the former method must be considered insufficient and not permanently tenable, for it permits of deductions only to a limited extent, and does not make it possible to give reasons for existing phenomena.

The remodelling which has become necessary requires undisturbed adherence to clear, simple, logical principles. What, however, is to be drawn from our criticism of the system hitherto used,—what I have endeavoured to illustrate and develope by single instances,—what the philosophical sentences I have quoted bring before us in a condensed form,—we may contract into one word. So far as our special problem is concerned, the question is to make the science of machinery deductive. The study must be so formed that it rests upon a few fundamental truths peculiar to itself. The whole fabric must be reducible to their strictness and simplicity, and from them again we must be able, conversely, to develope it. 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; indeed it has so often cried Eureka that it now rests quietly in the impression that some such fixed standpoint has really been found.

In the development of every exact science, its substance having grown sufficiently to make generalisation possible, there is a time when a series of changes brings it into clearness. This time has most certainly arrived for the science of Kinematics. The number of mechanisms has grown almost out of measure, and the number of ways in which they are applied no less. It has become absolutely impossible still to hold the thread which can lead in any way through this labyrinth by the existing methods.

It cannot be denied that the difficulties in the way of remodelling the science are great. We often do not know ourselves how closely wedged-in our ideas are by the boundaries which education and study have drawn around us. If new ideas therefore are to be substituted for the common ones, it can only be by a wrench sufficiently powerful to overcome the cohesion of established notions and prejudices. There are the traditional courses of instruction in the schools,—the widely extended and important technical literature,—the force of habit, acquired with difficulty, and on that very account firmly rooted; there is also the real difficulty that the new study requires to be grasped as a whole, and not taken up partially and occasionally,—all these pile up mighty hindrances. I cannot therefore shorten the way, although the truths to which it leads are of great simplicity. The careful removal of preconceived ideas, the slow seeking of the right path among those inviting us, prevents rapid motion. The following chapters are therefore intended not so much to add to the positive knowledge of the mechanician as to increase his understanding of what he already knows, so that it may become more thoroughly his own property. For, to conclude in the words of Göthe, "What is not understood is not possessed."

Note.—Had Prof. Reuleaux been acquainted with Prof. Rankine's "Machinery and Millwork," the first edition of which was published in 1869, he would no doubt have mentioned it here. Some 300 pages of it are devoted to Machine-Kinematics, or as Prof. Rankine calls it, the Geometry of Machinery,—and this subject is treated in a way which has some points in common with that now adopted by Reuleaux, although greatly differing from it.

Although neither Rankine's nomenclature nor his classification (given first partly in his Applied Mechanics) is now likely to be followed—it may be interesting to compare them with those of Reuleaux, the superiority of which I believe Rankine would have been the first to recognise had he lived to know them. Rankine considers a machine to be made up of a "frame" and "moving pieces" the latter being "primary" and "secondary." The frame is the fixed link of the mechanism (in the language of Reuleaux), the primary moving pieces are links the nature of whose motions are determined solely by their connection with the frame, the secondary moving pieces are all other links. He considers (erroneously) that the primary pieces can have no other motions than those he calls shifting, turning and helical, and that they must be connected to the frame by one of the lower pairs of elements. He then goes on to examine the simpler conditions of these three kinds of motion, which he does by the ordinary geometrical methods. The general nature of the motion of secondary moving pieces is treated by itself, principally by the method of instantaneous centres and axes, which Rankine afterwards uses freely and with great advantage throughout the work. In the next chapters, unfortunately, an entirely different set of ideas is introduced, and the comparative distinctness, as to system, of the earlier part of the work is lost. The idea of "elementary combinations" is brought in, and under this head are treated an immense number of kinematic chains of the most various descriptions, as well as the delineation of the profiles of several higher elements. The "mechanical powers" are placed as a subdivision under "elementary combinations." The lever and wheel and axle are taken together as cases of motion about a point, i.e. of turning,—and the inclined plane, wedge and screw as cases of sliding,—the "powers" are therefore not considered to be connected with the three simple motions of the first chapter. The place of the pulley among them is left unexplained. Some compound chains and some simple ones which in no way differ from "elementary combinations" are treated in two further chapters on "aggregate combinations" and "adjustments."

Notwithstanding the excellence of Prof. Rankine's book—and the value of some parts of it will be increased rather than diminished when read in the fresh light of Reuleaux's investigations,—it must be confessed that it contains neither a general theory of machines nor a systematic treatment of their motions. Most of its solutions are special rather than general, fresh methods being adopted for each new class of mechanisms. The real points of connection and of difference between them are thus lost, and the subject presents itself to the student as a series—or rather as many series—of interesting problems, which have only a very indistinct relation to each other. He is not led to any standpoint from which he can take a general survey of the whole. For this purpose the apparently simple, but in their results most wonderful ideas of the nature, pairing and inversion of machine-elements and kinematic chains were essential. Rankine unfortunately had no opportunity of making use of these; he does not seem either to have recognised the use of centroids, or the possibility of constrained motion in the higher pairs. He does not even point out that two of the motions of his "primary" pieces are in reality only special cases of the third, the twist, or that the method of instantaneous centres is as applicable to these motions as to more complicated ones. If it had not been for my own feeling as to the value of Prof. Rankine's work, I should not have alluded at such length to what appear to me to be defects or omissions in it. He has done great and lasting service in connection with the scientific study of machinery in this country, and his "Machinery and Mill work" is so familiar to engineers that it appeared to me impossible to leave unnoticed the bearing upon it of Reuleaux's Theory of Machines.

  1. Facsimile from Watt's letter. See Muirhead's Mechanical Inventions of James Watt, vol. ii. p. 88.
  2. See Muirhead as above, vol. ii. p. 93,—where also the Specification is given at full length with the necessary engravings.
  3. See Prof. Reuleaux's Geschichte der Dampfmaschine, Brunswick, 1864.
  4. A third edition of this appeared in 1840: it is a repetition of the second, slightly enlarged and got up in a better form.
  5. This was written before the death of Poncelet.
  6. Since the first publication of these remarks the second edition of Prof. Willis' work has appeared. (London 1870.) It is considerably in advance of the first, but in all essential points the principles originally laid down are unchanged. It so far confirms my view that the author, while retaining his original divisions and subdivisions, has inverted their order. R.
  7. The work of Prof. Rankine in connection with Machine-Kinematics, which was in some respects remarkable, I have mentioned in a note at the end of the Introduction.