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It is much to be regretted that this distinguished writer did not attempt to follow up this clear and able view of the subject, by actually developing the science in question.
A similar separation of the principles of motion and force formed the basis of the Lectures on Mechanism which I delivered for the first time to the University of Cambridge, in 1837; and the same views were subsequently sanctioned by the high authority of Professor Whewell, who, in his Philosophy of the Inductive Sciences, has assigned a chapter to the Doctrine of Motion*, in which, under the title of Pure Mechanism, he has defined this science nearly in the above words of Ampère, whom he quotes.
To make the plan of the following pages more intelligible, it will be necessary in the first place to take a short review of the system of MM. Lanz and Betancourt, which, as we have seen, is founded upon the views of Monge. Their system is thus detailed at the opening of their work :
“ The motions of the parts of machines are either (1) Rectilinear, (2) circular, (3) or curvilinear; and each of these may be continuous in direction or alternate, that is, back and forward. These six motions admit of being combined two and two in twenty-one different ways, each motion being supposed to be also combined with itself. The object of every simple machine being to counterchange or communicate these motions, the following system will include them all.
* Whewell, Philosophy of the Inductive Sciences, 1840, p. 144.
continuoust 1 alternatet 2
continuoust 3 Continuous Rectilinear*, changed into circular...
curvilinear continuoust 5
alternatet rectilinear alternatet
circular... Scontinuoust §
Continuous Circular*, into
continuoust 10 curvilinear
alternatet 11 (rectilinear alternatet 12 circular... alternatet
13 Continuous Curvilinear*, into
continuoust 14 curvilinear alternatet
rectilinear alternatet 16 Alternate Rectilinear, into
circular... alternatet 17
19 Alternate Circular*, into.....
curvilinear alternatet 20 Alternate Curvilinear*, into.... curvilinear alternatet 21"
Teden seit 30?
7 Of many of these combinations, however, no direct solu. tion is given. Thus for (2) we are told to convert rectilinear motion into circular by one of the combinations in (3), and then to convert this into alternate rectilinear by one of those in (7). In this way also classes 5, 6, 11, 12, 13, 15, 16, 18, and 21, are disposed of ; so that there remain only twelve, under which our authors proceed to arrange the elementary combinations into which, according to them, mechanism may be resolved.
This celebrated system, which has been pretty generally received, must however be considered as a merely popular arrangement, notwithstanding the apparently scientific sim
* With velocity either uniform or varying according to a given law.
+ With a velocity of the same nature as that which produces it, preserving a constant proportion to it or varying according to a given law. In the same or in different planes.
plicity of the scheme. In the first place, it is not confined to pure combinations of mechanism, but is embarrassed by the intrusion of several dynamical and even hydraulic contrivances. Thus, a water-wheel and a windmill-sail are considered to be a means of converting continuous rectilinear motion into continuous circular; and a ferry-boat attached to one end of a long rope, of which the other is fixed to the bank, is admitted into Class 4, as a means of converting continuous rectilinear motion into alternate circular. Flywheels, pendulums with their escapements, parallel motions, are all placed in one class or other of this scheme. No attempt is made to subject the motions to calculation, or to reduce these laws to general formulæ, for which indeed the system is totally unfitted.
The plan of the great work of Borgnis, published in 1818, is much more comprehensive and complete, really embracing the whole subject of machinery, instead of being confined by its plan to elementary combinations for the modification of motion. Borgnis, in the volume on the Composition of Machines, divides mechanical organs into six orders, each of which have subordinate classes. His orders are *; (1) Receivers of power; (2) Communicators; (3) Modifiers; (4) Frame-work, fixed and moveable : (5) Regulators; (6) Working parts.
For the mere purposes of descriptive mechanism this system is much better adapted than that of MM. Lanz
* In the original, (1) Récepteurs, (2) Communicateurs, (3) Modificateurs, (4) Supports, (5) Régulateurs, (6) Opératenrs.
and Betancourt, but still does not provide for the investigation of the laws of the modifications of motion, which is an especial object of the proposed science of Kinematics. Many essays, however, have been from time to time written concerning various detached portions of this science. The teeth of wheels is the most remarkable of these, from having occupied the attention of so many of the best mathematicians. But in fact, the description of all the mechanical curves, as epicycloids and conchoids, may be held to belong to this science, which would thus be made to include a great mass of matter that has hitherto been classed with geometry. The calculation of trains of wheel-work is also a branch of it, to which the first contribution was made by Huyghens, who employed continued fractions, for the purpose of obtaining approximate numbers for the trains of his Planetarium *.
The following pages must not however be considered as an attempt to carry out the able and comprehensive views of Ampère, being confined to machinery alone, and not passing from it to the more abstract generalities of motion, which he seems to have contemplated.
My object has been to form a system that would embrace all the elementary combinations of mechanism, and at the same time admit of a mathematical investigation of the laws by which their modifications of motion are governed. I have confined myself to the Elements of Pure
Vide also Young's Nat. Philosophy, vol. 11. p. 55. Arts. 365, 366, the substance of which will be found in this work. Arts. 34 and 237.
Mechanism, that is, to those contrivances by which motion is communicated purely by the connexion of parts, without requiring the essential intermixture of dynamical effects.
I have taken a different course from the one hitherto followed, in respect that instead of considering a machine to be an instrument by means of which we may change the direction and velocity of a given motion, I have treated it as an instrument by means of which we may produce any relations of motion between two pieces.
For Monge and his followers began by dividing motion into rectilinear and rotative, continuous and reciprocating, and so based their system upon the actual motion of the parts; and Ampère defines his machine in the words quoted above as modifying a given motion. But a little consideration will shew that any given element of machinery can only govern the relations of velocity and direction of the pieces it serves to connect; and that this connexion and the law of its action are for the most part independent of the actual velocities. By establishing a system upon the relations of motion instead of upon the actual motions, it will be found that many of the redundancies and difficulties that have hitherto obscured the subject are got rid of.
Thus, to follow up the example given by Ampère of the hands of a watch, it is clear that the connexion governs the relation of their angular velocities, which at every instant is in the proportion of twelve to one; and also provides that