These hypothetical conditions would find themselves exactly realized in the highest degree in _chemical_ phenomena, if we attempted to calculate them by the theory of general gravitation. On properly weighing the preceding considerations, the reader will be convinced, I think, that in reducing the future extension of the great applications of mathematical analysis, which are really possible, to the field comprised in the different departments of inorganic physics, I have rather exaggerated than contracted the extent of its actual domain. Important as it was to render apparent the rigorous logical universality of mathematical science, it was equally so to indicate the conditions which limit for us its real extension, so as not to contribute to lead the human mind astray from the true scientific direction in the study of the most complicated phenomena, by the chimerical search after an impossible perfection. * * * * * Having thus exhibited the essential object and the principal composition of mathematical science, as well as its general relations with the whole body of natural philosophy, we have now to pass to the special examination of the great sciences of which it is composed. _Note._--ANALYSIS and GEOMETRY are the two great heads under which the subject is about to be examined. To these _M. Comte_ adds Rational MECHANICS; but as it is not comprised in the usual idea of Mathematics, and as its discussion would be of but limited utility and interest, it is not included in the present translation. BOOK I. ANALYSIS. BOOK I. ANALYSIS. CHAPTER I. GENERAL VIEW OF MATHEMATICAL ANALYSIS.