In order to do this, we must reconsider, in a more definite and precise shape, the doctrine which has already been laid down;--that our Knowledge consists {29} in applying Ideas to Facts; and that the conditions of real knowledge are that the ideas be distinct and appropriate, and exactly applied to clear and certain facts. The steps by which our knowledge is advanced are those by which one or the other of these two processes is rendered more complete;--by which _Conceptions_ are _made more clear_ in themselves, or by which the Conceptions more strictly _bind together the Facts_. These two processes may be considered as together constituting the whole formation of our knowledge; and the principles which have been established in the History of Scientific Ideas bear principally upon the former of these two operations;--upon the business of elevating our conceptions to the highest possible point of precision and generality. But these two portions of the progress of knowledge are so clearly connected with each other, that we shall deal with them in immediate succession. And having now to consider these operations in a more exact and formal manner than it was before possible to do, we shall designate them by certain constant and technical phrases. We shall speak of the two processes by which we arrive at science, as _the Explication of Conceptions_ and _the Colligation of Facts_: we shall show how the discussions in which we have been engaged have been necessary in order to promote the former of these offices; and we shall endeavour to point out modes, maxims, and principles by which the second of the two tasks may also be furthered. {{30}} CHAPTER II. OF THE EXPLICATION OF CONCEPTIONS. APHORISM II. _The Explication of Conceptions, as requisite for the progress of science, has been effected by means of discussions and controversies among scientists; often by debates concerning definitions; these controversies have frequently led to the establishment of a Definition; but along with the Definition, a corresponding Proposition has always been expressed or implied. The essential requisite for the advance of science is the clearness of the Conception, not the establishment of a Definition. The construction of an exact Definition is often very difficult. The requisite conditions of clear Conceptions may often be expressed by Axioms as well as by Definitions._ APHORISM III. _Conceptions, for purposes of science, must be_ appropriate _as well as clear: that is, they must be modifications of_ that _Fundamental Idea, by which the phenomena can really be interpreted. This maxim may warn us from errour, though it may not lead to discovery. Discovery depends upon the previous cultivation or natural clearness of the appropriate Idea, and therefore_ no discovery is the work of accident. SECT. I.--_Historical Progress of the Explication of Conceptions._