Since it is impossible for different numbers to have any of their modes of formation completely in common, it is a kind of paradox to say, that all propositions which can be made concerning numbers relate to their modes of formation from other numbers, and yet that there are propositions which are true of all numbers. But this very paradox leads to the real principle of generalization concerning the properties of numbers. Two different numbers cannot be formed in the same manner from the same numbers; but they may be formed in the same manner from different numbers; as nine is formed from three by multiplying it into itself, and sixteen is formed from four by the same process. Thus there arises a classification of modes of formation, or in the language commonly used by mathematicians, a classification of Functions. Any number, considered as formed from any other number, is called a function of it; and there are as many kinds of functions as there are modes of formation. The simple functions are by no means numerous, most functions being formed by the combination of several of the operations which form simple functions, or by successive repetitions of some one of those operations. The simple functions of any number _x_ are all reducible to the following forms: _x + a_, _x - a_, _a x_, _x/a_, _x^a_, _a [root of] x_, log. _x_ (to the base _a_), and the same expressions varied by putting _x_ for _a_ and _a_ for _x_, wherever that substitution would alter the value: to which perhaps ought to be added sin _x_, and arc (sin = _x_). All other functions of _x_ are formed by putting some one or more of the simple functions in the place of _x_ or _a_, and subjecting them to the same elementary operations. In order to carry on general reasonings on the subject of Functions, we require a nomenclature enabling us to express any two numbers by names which, without specifying what particular numbers they are, shall show what function each is of the other; or, in other words, shall put in evidence their mode of formation from one another. The system of general language called algebraical notation does this. The expressions _a_ and _a^2 + 3a_ denote, the one any number, the other the number formed from it in a particular manner. The expressions _a_, _b_, _n_, and _(a + b)^n_, denote any three numbers, and a fourth which is formed from them in a certain mode.