We are now to enter upon the study of the first of the Six great Sciences : and we begin by establishing the importance of the Positive Philosophy in perfecting the character of each science in itself.

Though Mathematics is the most ancient and the most perfect science of all, the general idea of it is far from being clearly determined. The definition of the science, and its chief divisions, have remained up to this time vague and uncertain. The plural form of the name (grammatically used as singular) indicates the want of unity in its philosophical character, as commonly conceived. In fact, it is only since the beginning of the last century that it could be conceived of as a whole; and since that time geometers have been too much engaged on its different branches, and in applying it to the most important laws of the universe, to have much attention left for the general system of the science. Now however the pursuit of its specialities is no longer so engrossing as to exclude us from the study of Mathematics in its unity. It has now reached a degree of consistency Avhich admits of the effort to reduce its parts into a system, in preparation for further advance. The latest achievements of mathematicians have prepared the way for this by evidencing a character of unity in its principal parts which was not before known to exist.