3. (II.) _Idea of Number, &c._--The study of mathematics, as I need hardly observe, developes and renders exact, our conceptions of the relations of number, as well as of space. And although, as we have already noticed, even in their original form the conceptions of number are for the most part very distinct, they may be still further improved by such discipline. In complex cases, a methodical cultivation of the mind in such subjects is needed: for instance, questions concerning Cycles, and Intercalations, and Epacts, and the like, require very great steadiness of arithmetical apprehension in order that the reasoner may deal with them rightly. In the same manner, a mastery of problems belonging to the science of Pure Motion, or, as I have termed it, _Mechanism_, requires either great natural aptitude in the student, or a mind properly disciplined by suitable branches of mathematical study. 4. Arithmetic and Geometry have long been standard portions of the education of cultured persons throughout the civilized world; and hence all such persons have been able to accept and comprehend those portions of science which depend upon the idea of space: for instance, the doctrine of the globular form of the earth, with its consequences, such as the measures of latitude and longitude;--the heliocentric system of the universe in modern, or the geocentric in ancient times;--the explanation of the rainbow; and the like. In nations where there is no such education, these portions of science cannot exist as a part of the general stock of the knowledge of society, however intelligently they {168} may be pursued by single philosophers dispersed here and there in the community.