is a principle of the greatest importance in the formation of mathematical science, according to the wide generality which such science has in modern times assumed. 3. _Connexion of Symbols and Analysis._--Since in our symbolical reasoning our symbols thus reason for us, we do not necessarily here, as in geometrical reasoning, go on adding carefully one known truth to another, till we reach the desired result. On the contrary, if we have a theorem to prove or a problem to solve which can be brought under the domain of our symbols, we may at once state the given but unproved truth, or the given combination of unknown quantities, in its symbolical form. After this first process, we may then proceed to trace, by means of our symbols, what other truth is involved in the one just stated, or what the unknown symbols must signify; resolving step by step the symbolical assertion with which we began, into others more fitted for our purpose. The former process is a kind of _synthesis_, the latter is termed _analysis_. And although symbolical reasoning does not necessarily imply such analysis; yet the connexion is so familiar, that the term _analysis_ is frequently used to designate symbolical reasoning. {{156}} CHAPTER XIII. THE DOCTRINE OF MOTION. 1. _Pure Mechanism._--THE doctrine of Motion, of which we have here to speak, is that in which motion is considered quite independently of its cause, force; for all consideration of force belongs to a class of ideas entirely different from those with which we are here concerned. In this view it may be termed the _pure_ doctrine of motion, since it has to do solely with space and time, which are the subjects of pure mathematics. (See c. i. of this book.) Although the doctrine of motion in connexion with force, which is the subject of mechanics, is by far the most important form in which the consideration of motion enters into the formation of our sciences, the Pure Doctrine of Motion, which treats of space, time, and velocity, might be followed out so as to give rise to a very considerable and curious body of science. Such a science is the science of Mechanism, independent of force, and considered as the solution of a problem which may be thus enunciated: 'To communicate any given motion from a first mover to a given body.' The science which should have for its object to solve all the various cases into which this problem would ramify, might be termed _Pure Mechanism_, in contradistinction to _Mechanics Proper_, or _Machinery_, in which Force is taken into consideration. The greater part of the machines which have been constructed for use in manufactures have been practical solutions of some of the cases of this problem. We have also important contributions to such a science in the works of Mathematicians; for example, the various investigations and demonstrations which have been published respecting the form of the Teeth {157} of Wheels, and Mr. Babbage's memoir[15\2] on the Language of Machinery. There