The nature of abstract mathematics (the general division of which will be examined in the following chapter) is clearly and exactly determined. It is composed of what is called the _Calculus_,[2] taking this word in its greatest extent, which reaches from the most simple numerical operations to the most sublime combinations of transcendental analysis. The _Calculus_ has the solution of all questions relating to numbers for its peculiar object. Its _starting point_ is, constantly and necessarily, the knowledge of the precise relations, _i.e._, of the _equations_, between the different magnitudes which are simultaneously considered; that which is, on the contrary, the _stopping point_ of concrete mathematics. However complicated, or however indirect these relations may be, the final object of the calculus always is to obtain from them the values of the unknown quantities by means of those which are known. This _science_, although nearer perfection than any other, is really little advanced as yet, so that this object is rarely attained in a manner completely satisfactory. [Footnote 2: The translator has felt justified in employing this very convenient word (for which our language has no precise equivalent) as an English one, in its most extended sense, in spite of its being often popularly confounded with its Differential and Integral department.] Mathematical analysis is, then, the true rational basis of the entire system of our actual knowledge. It constitutes the first and the most perfect of all the fundamental sciences. The ideas with which it occupies itself are the most universal, the most abstract, and the most simple which it is possible for us to conceive.