The greater part of the Theoretische Kinematik of Prof. Reuleaux, which I have now the pleasure of presenting to English and American leaders, was originally published in chapters in the Berliner Verhandlungen, under the title of Kinematische Mittheilungen, These papers, revised and enlarged, and with the addition of a chapter on Kinematic Synthesis, were published collectively in 1874-5 in the work of which the present is a translation. They have attracted considerable attention in Germany, and the principles laid down in them have already made their way into Polytechnic School instruction, not only in that country but also in Russia and Italy.

The book addresses itself to somewhat different classes of readers, or rather to readers who have had very different training, on the Continent and here. Its readers there are to a great extent the past or present students of the Polytechnic Schools, or at least those who are acquainted with polytechnic teaching. They are familiar with a regularly systematised system of machine instruction and its somewhat extended literature. Here, on the other hand, neither systematised instruction nor extended literature exists. The book addresses itself greatly to practical engineers and mechanicians, men who have often enough worked out their knowledge of the subject for themselves to a far greater extent than they have acquired it from books or lectures. To these readers some sections of the book may appear unnecessary, as referring to opinions or combating conclusions of which they have scarcely heard, and the erroneousness of which they are perfectly ready to admit. No doubt had the work originally been written for its English readers these passages might have been omitted or changed; as it is, I must merely remind those readers of the fact I have just mentioned. Here and there I have made small alterations in the text on this account, otherwise the sections referred to remain as in the original. The conclusions arrived at in them are not the less interesting that they might have been reached here, sometimes, in a more direct manner.

It may be well for me to mention here some of the leading characteristics of Prof. Reuleaux's treatment of his subject, and to point out in what respects it differs from that of his predecessors. In the oldest books upon machinery each machine was taken up as a whole, to be described and treated by itself from beginning to end. Gradually it became recognised that similar parts occurred again and again in different machines, and these parts received the name of mechanisms. They sometimes appear in a more or less abstract form in text-books of Elementary Mechanics, and have received more complete treatment in separate works. With the growth of clear ideas in physical science it became possible to separate the ideas of force, time and motion, and to consider the latter merely for its own sake without reference to the other two. Prof. Willis adopted this treatment unreservedly in his Principles of Mechanism - a work too well known to need any characterisation here - calling the study thus marked out the "Science of Pure Mechanism." Here, however, the matter stopped, later writers have been content to follow upon Willis's lines, not carrying the analytic process further, and con- tenting themselves with the examination of mechanisms as a whole in the forms in which they are presented to us by tradition or invention, without attempting to analyse them, or to investigate their mode of formation.

It is unquestionably true that by the aid of mathematics this treatment of mechanisms has given us many most valuable results, but it is equally true that the method itself is defective, and was only used for want of a better. This better method Prof. Reuleaux has attempted, and I think with great success, to indicate. Starting with the idea of motion as change of position only - and limiting himself to cases where such changes are absolutely determinate at every instant, - as always in the machine - he points out that they are conditioned simply by the geometric form of the moving bodies. Two bodies, such for instance as a screw and nut, having such forms that at any instant there is only one possible motion for each relatively to the other, form the simplest combination available for machinal purposes - such bodies he calls a pair of elements. Two or more elements from as many different pairs can be combined into a link, and such links united into kinematic chains, and it is by fixing, that is, preventing the motion of, someone link of such a chain that a mechanism is obtained. Stated thus in a few words the analysis is simple and obvious enough; like many other simple things, however, it leads to most important consequences. As one illustration merely of this, I may point to the collection of “rotary” engines and pumps examined in Chap. IX. Here will be found, among others, over thirty forms of "rotary" engines of which the kinematic chain used in the driving mechanism is absolutely identical with that of the common direct-acting engine! Their constructive forms differ most widely, and have of course too often misled their inventors, but the application of what Prof. Reuleaux calls “kinematic analysis” shows at once both their identity as kinematic chains and their relation as mechanisms. In Fig. 3, PI. XX., for instance, is shown a rotary engine which has been patented every few years since 1805 in one or another form, and in which no doubt some of my readers will recognise an old friend "schemed" in the days of their apprenticeship. Its driving mechanism is absolutely the same as that of the direct-acting engine, but with the crank fixed and the frame allowed to move round it.

In order to utilise the kinematic analysis Prof. Reuleaux has devised and elaborated the notation which is explained in Chap. VII. and used in the later part of the book. That this notation is both exceedingly simple and of practical use will be admitted by all readers of Chap. IX., but its full advantages will only be realised by those who use it for themselves. The way in which it aids the resolution of apparently complex mechanisms into quite familiar forms is often most remarkable. Use will no doubt suggest modifications and improvements in its details, but Prof. Reuleaux is very anxious that its essential features, and especially the symbols for the elements (which have been so chosen as to be as suitable as possible for the principal European languages) should not be altered.

I may mention here only one other feature in Reuleaux's work, namely, his treatment of fluids when they occur in mechanisms or machines (Chap. IV. &c.). It has long been customary, of course, to treat cords, chains, belts &c., as organs which could legitimately form part of machines, but fluids have been universally (so far as I know) excluded from consideration in this way. Reuleaux points out that fluids - "pressure-organs" - are simply contrapositives of the "tension-organs" just mentioned, and that if one be included in the study of "pure mechanism" there can be no reason for excluding the other. He gives also many instances of the way in which engineers use the one or the other as the column of fluid or the cord best suits their purpose. In examining mechanisms we consider the motions of each body as a whole, ignoring altogether its molecular condition, or more strictly assuming that it is so arranged that its molecular stability is not disturbed during the motion. This pre-supposition is made tacitly in the case of "rigid" bodies, where molecular stability is independent of the application of external force. It is made also in the case of ropes, belts, &c., for when these occur in machines it is always assumed that they are kept in tension by some force external to themselves, in any other case their motions would be quite indeterminate. With fluids it is not necessary to make any other assumption than this, but the external force must be a pressure instead of a pull, and must be supplied in directions other than that in which motion takes place

The scientific carrying-out in practice of the requirements covered by our definition of the machine has caused the rise of an extended apparatus of sciences in connection with the progressing development of Polytechnic instruction. From the foundation sciences of Mathematics and Physics three or four other sciences specially concerning the machine have separated themselves. Their common object is the elucidation of the causal-connection of machine phenomena. Together they have been happily enough called Practical Mechanics. I speak of them here as. sciences without pretending to insist on their absolute right to the title; they may be called sciences of the second or third order, or by their usual names; they employ scientific methods, and treat by their means special regions of investigation ; within these they have reached by degrees an independence which has made necessary their separation from the more general sciences.

First comes the study of machines in general, looked at in connection with the work they have to do. This is known in Germany under a number of somewhat vague titles, as general or descriptive, special and theoretical "Maschinenlehre" In its general form it deals, descriptively, with the whole of existing machines, it teaches what machines exist and how they are constituted, and thus affords us a glance at their manner of growth. It proceeds teleologically in the fullest sense of the word, seeking always to refer everything to the special object for which the machine was constructed. Its methods of classification are made as general as possible. At present a complete descriptive, or really general treatment of machinery in this way, is hardly possible on account of the enormous number of existing machines. To be really general only classes and types can be treated of quietly adapting itself to the every-day wants of the learner the study thus becomes specialised, single classes are taken up and treated singly in detail. Along with the construction of each special machine its theory is also, for the most part, considered, that is, the nature of the sensible forces which come into action and the motions to which they give rise are examined, and deductions are drawn m regard to the most suitable way for turning these forces to account. This method of treatment is therefore based also on existing machines, but differs from the former in not only describing their arrangement and purpose, but in examining also how they can best be arranged in order to carry out the given purpose. In Germany at present it is for the most part rightly grasped and comprehended, the machine itself being taken as both the end and the beginning of the problem. The French, however, have not always freed themselves from the idea that the machine occurs merely as an illustration an example in Applied Mechanics; if this idea were right, however, it is clear that all other applications of Mechanics should be treated in the same way. If, coming somewhat nearer to the heart of tie matter, the applications of mechanics "in machinery" be classed by themselves, as is done by Poncelet, still the principle is not carried sufficiently far, for under this title all machines of every kind must be treated, which, however, has not been the case. Redtenbacher first removed this stigma of indistinctness from the matter, and thereby laid the foundation of the freshness and power which the German system of machine-instruction shows as contrasted with the French. Redtenbacher's most lasting services, which have not always been understood by his successors, lie in this direction, in the separation of the questions connected with machinery into separate sciences or branches of science. It was on this ground that his influence was, I may say, so electric, and brought to him so quickly in his time the engineering students of Germany.

The existing treatment of the theory of machines (theoretische Maschinenlehre) confines itself principally to prime-movers, Steam-engines, Water-wheels, Turbines, Windmills, and so on, or in terms of our definition, it concerns itself with the nature of the various arrangements by means of which natural forces can be best applied in machinery. Yet it does also consider machines in general (other than prime-movers), and obviously these all belong to its province. To the general examination of the theory of these machines the name mechanical technology is often given. This is not universal, nor indeed is it correct, for mechanical technology must include all mechanical processes of manufacture, and in a multitude of cases machines are not employed in these. It possesses therefore a domain of its own, and must be treated in its own proper way. From its own point of view it also examines the machine, but in a way entirely differing from that in which it is examined for its own sake in the studies of which we are speaking. While therefore it can easily be understood how both studies should set up claims to the same object of instruction, it is on that very account important that they should not be confused with each other.

The special part of technology here coming into question, or what may be called the technological part of the study of special machines, concerns itself with the action of the natural forces, through their various applications in the machine, on the bodies to be worked upon. It examines, in other words, by what special arrangement of the parts of the machine the required action can be best obtained. As a whole, therefore, the specialised study of machines (specielle Maschinenlehre) considers both the application of the natural forces to a given machine and their action in it.

The third science is that of Machine-design. It also has been freed by Redtenbacher from its incorrect treatment under Applied Mechanics, and placed by him on an independent footing. Its province is to teach how to give to the bodies constituting the machine the capacity for resisting alteration of form mentioned in our definition. In order to determine this property fully it must be considered in reference not only to sensible but also to latent forces.