I shall have occasion to cite presently, for another reason, a new example, very suitable to make apparent the fundamental distinction which I have just exhibited; it is that of circular functions, both direct and inverse, which at the present time are still sometimes concrete, sometimes abstract, according to the point of view under which they are regarded.
À posteriori, the general character which renders a function abstract or concrete having been established, the question as to whether a certain determinate function is veritably abstract, and therefore susceptible of entering into true analytical equations, becomes a simple question of fact, inasmuch as we are going to enumerate all the functions of this species.
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 = [aroot]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] |
Such are the elements, very few in number, which directly compose all the abstract functions known at the present day. Few as they are, they are evidently sufficient to give rise to an infinite number of analytical combinations.
No rational consideration rigorously circumscribes, à priori, the preceding table, which is only the actual expression of the present state of the science. Our analytical elements are at the present day more numerous than