_The Perception of Time involves a constant and latent kind of memory, which may be termed a_ Sense of Succession. _The Perception of Number also involves this Sense of Succession, although in small numbers we appear to apprehend the units simultaneously and not successively._ (II. 10.) XXXVIII. _The Perception of Rhythm is not an impression on the passive sense, but requires an act of thought by which we connect and group the strokes which form the Rhythm._ (II. 10.) XXXIX. Intuitive _is opposed to_ Discursive _reason. In intuition, we obtain our conclusions by dwelling upon_ one _aspect of the fundamental Idea; in discursive reasoning, we combine_ several _aspects of the Idea,_ (_that is, several axioms,_) _and reason from the combination._ (II. 11.) XL. _Geometrical deduction_ (_and deduction in general_) _is called_ Synthesis, _because we introduce, at successive steps, the_ {12} _results of new principles. But in reasoning on the relations of space, we sometimes go on_ separating _truths into their component truths, and these into other component truths; and so on: and this is geometrical_ Analysis. (II. 11.) XLI. _Among the foundations of the Higher Mathematics, is the_ Idea of Symbols _considered as general_ Signs _of Quantity. This idea of a Sign is distinct from, and independent of other ideas. The Axiom to which we refer in reasoning by means of Symbols of quantity is this_:--The interpretation of such symbols must be perfectly general. _This Idea **and Axiom are the bases of Algebra in its most general form._ (II. 12.) XLII. _Among the foundations of the Higher Mathematics is also the_ Idea of a Limit. _The Idea of a Limit cannot be superseded by any other definitions or Hypotheses, The Axiom which we employ in introducing this Idea into our reasoning is this_:--What is true up to the Limit is true at the Limit. _This Idea and Axiom are the bases of all Methods of Limits, Fluxions, Differentials, Variations, and the like._ (II. 12.) XLIII. _There is a_ pure _Science of Motion, which does not depend upon observed facts, but upon the Idea of motion. It may also be termed_ Pure Mechanism, _in opposition to Mechanics Proper, or_ Machinery, _which involves the mechanical conceptions of force and matter. It has been proposed to name this Pure Science of Motion,_ Kinematics. (II. 13.) XLIV. _The pure Mathematical Sciences must be successfully cultivated, in order that the progress of the principal Inductive Sciences may take place. This appears in the case of Astronomy, in which Science, both in ancient and in modern times, each advance of the theory has depended upon the_ {13} _previous solution of problems in pure mathematics. It appears also inversely in the Science of the Tides, in which, at present, we cannot advance in the theory, because we cannot solve the requisite problems in the Integral Calculus._ (II. 14.) XLV.