11. Our Conceptions, then, even when they become so clear as the progress of knowledge requires, are not adequately expressed, or necessarily expressed at all, by means of Definitions. We may ask, then, whether there is any _other mode_ of expression in which we may look for the evidence and exposition of that peculiar exactness of thought which the formation of Science demands. And in answer to this inquiry, we may refer to the discussions respecting many of the Fundamental Ideas of the sciences contained in our _History_ of such Ideas. It has there been seen that these Ideas involve many elementary truths which enter into the texture of our knowledge, introducing into it connexions and relations of the most important kind, although these elementary truths cannot be deduced from any verbal definition of the idea. It has been seen that these elementary truths may often be enunciated by means of _Axioms_, stated in addition to, or in preference to, Definitions. For example, the Idea of Cause, which forms the basis of the science of Mechanics, makes its appearance in our elementary mechanical reasonings, not as a Definition, but by means of the Axioms that 'Causes are measured by their effects,' and that 'Reaction is equal and opposite to action.' Such axioms, tacitly assumed or {41} occasionally stated, as maxims of acknowledged validity, belong to all the Ideas which form the foundations of the sciences, and are constantly employed in the reasoning and speculations of those who think clearly on such subjects. It may often be a task of some difficulty to detect and enunciate in words the Principles which are thus, perhaps silently and unconsciously, taken for granted by those who have a share in the establishment of scientific truth: but inasmuch as these Principles are an essential element in our knowledge, it is very important to our present purpose to separate them from the associated materials, and to trace them to their origin. This accordingly I attempted to do, with regard to a considerable number of the most prominent of such Ideas, in the _History_. The reader will there find many of these Ideas resolved into Axioms and Principles by means of which their effect upon the elementary reasonings of the various sciences may be expressed. That Work is intended to form, in some measure, a representation of the Ideal Side of our physical knowledge;--a Table of those contents of our Conceptions which are not received directly from facts;--an exhibition of Rules to which we know that truth must conform. SECT. IV.--_Clear and appropriate Ideas._