_Enumeration of Abstract Functions._ At first view this enumeration seems impossible, the distinct analytical functions being infinite in number. But when we divide them into _simple_ and _compound_, the difficulty disappears; for, though the number of the different functions considered in mathematical analysis is really infinite, they are, on the contrary, even at the present day, composed of a very small number of elementary functions, which can be easily assigned, and which are evidently sufficient for deciding the abstract or concrete character of any given function; which will be of the one or the other nature, according as it shall be composed exclusively of these simple abstract functions, or as it shall include others. We evidently have to consider, for this purpose, only the functions of a single variable, since those relative to several independent variables are constantly, by their nature, more or less _compound_. Let _x_ be the independent variable, _y_ the correlative variable which depends upon it. The different simple modes of abstract dependence, which we can now conceive between _y_ and _x_, are expressed by the ten following elementary formulas, in which each function is coupled with its _inverse_, that is, with that which would be obtained from the direct function by referring _x_ to _y_, instead of referring _y_ to _x_. FUNCTION. ITS NAME. 1st couple {1° _y_ = _a_ + _x_ _Sum._ {2° _y_ = _a_ - _x_ _Difference._ 2d couple {1° _y_ = _ax_ _Product._ {2° _y_ = _a/x_ _Quotient._ 3d couple {1° _y_ = _x^a_ _Power._ {2° _y_ = _[ath root]x_ _Root._ 4th couple {1° _y_ = _a^x_ _Exponential._ {2° _y_ = _[log a]x_ _Logarithmic._ 5th couple {1° _y_ = sin. _x_ _Direct Circular._ {2° _y_ = arc(sin. = _x_). _Inverse Circular._[3] [Footnote 3: With the view of increasing as much as possible the resources and the extent (now so insufficient) of mathematical analysis, geometers count this last couple of functions among the analytical elements. Although this inscription is strictly legitimate, it is important to remark that circular functions are not exactly in the same situation as the other abstract elementary functions. There is this very essential difference, that the functions of the four first couples are at the same time simple and abstract, while the circular functions, which may manifest each character in succession, according to the point of view under which they are considered and the manner in which they are employed, never present these two properties simultaneously.