_The Methods by which the acquisition of clear Scientific Ideas is promoted, are mainly two_; Intellectual Education _and_ Discussion of Ideas. APHORISM XXX. _The Idea of Space becomes more clear by studying_ Geometry; _the Idea of Force, by studying_ Mechanics; _the Ideas of Likeness, of Kind, of Subordination of Classes, by studying_ Natural History. APHORISM XXXI. Elementary Mechanics _should now form a part of intellectual education, in order that the student may understand the Theory of Universal Gravitation: for an intellectual education should cultivate such ideas as enable the student to understand the most complete and admirable portions of the knowledge which the human race has attained to._ APHORISM XXXII. Natural History _ought to form a part of intellectual education, in order to correct certain prejudices which arise from cultivating the intellect by means of mathematics alone; and in order to lead the student to see that the division of things into Kinds, and the attribution and use of Names, are processes susceptible of great precision._ {165} THE ways in which men become masters of those clear and yet comprehensive conceptions which the formation and reception of science require, are mainly two; which, although we cannot reduce them to any exact scheme, we may still, in a loose use of the term, call _Methods_ of acquiring clear Ideas. These two ways are Education and Discussion. 1. (I.) _Idea of Space._--It is easily seen that Education may do at least something to render our ideas distinct and precise. To learn Geometry in youth, tends, manifestly, to render our idea of space clear and exact. By such an education, all the relations, and all the consequences of this idea, come to be readily and steadily apprehended; and thus it becomes easy for us to understand portions of science which otherwise we should by no means be able to comprehend. The conception of _similar triangles_ was to be mastered, before the disciples of Thales could see the validity of his method of determining the height of lofty objects by the length of their shadows. The conception of _the sphere with its circles_ had to become familiar, before the annual motion of the sun and its influence upon the lengths of days could be rightly traced. The properties of circles, combined with the _pure_[13\3] _doctrine of motion_, were required as an introduction to the theory of Epicycles: the properties of _conic sections_ were needed, as a preparation for the discoveries of Kepler. And not only was it necessary that men should possess a _knowledge_ of certain figures and their properties; but it was equally necessary that they should have the _habit of reasoning_ with perfect steadiness, precision, and conclusiveness concerning the relations of space. No small discipline of the mind is requisite, in most cases, to accustom it to go, with complete insight and security, through the demonstrations respecting intersecting planes and lines, dihedral