VARIABLE. A quantity is said to vary when it changes value, whether gradually, or by jumps or starts. The notion of a variable quantity is the first which must be established in teaching the Diffe rential Calculus, and requires a little explanation.
One magnitude at least is hardly conceivable without the notion of variation ; we mean time or duration. Reckoning from a fixed epoch, the idea of the present time is nothing but that of the other extremity of a variable quantity, the variation of which we cannot suspend, even in thought. Again, in spaee.mag-nitudes, though we are not obliged to consider them as formed by variation, yet it is in our power to do so, and we are constantly learning the variation of length, area, or solidity consequent upon motion. And we can even consider this variation as arising from no act of our own, as independent of us, and out of our power to stop : though even when this is physically true, namely, that the variation is out of our power, we can conceive or imagine that it does stop, and trace the consequences of such stoppage. Variable magnitude, then, presents natural ideas, such as we not only easily acquire, but such as it would be difficult, if not impossible, to suppose that we could help acquiring.
But when we come to speak of number, the case is much altered. The constant phrase of an algebraist, "let x be a variable quantity," clear as it may be when quantity means magnitude, is not quite so plain when quantity means number as the representative of magnitude. There is something to be said as to how number is imagined to vary at all : and still more as to its gradual variation.
Number is an abstraction of the mind ; it is not magnitude, but a mode of reference of one magnitude to another. If we might dare to say it, number is more of the nature of an opinion about magnitude than of magnitude itself. When we speak of a symbol representing a variable number, we know that, though we say the symbol changes its values, it is we ourselves who arbitrarily change the meaning of the symbol. We can imagine (waving all question about the possibility of our imagination, or its metaphysical truth) everything annihilated except two material points, one or both of which are in motion with respect to the other : but we cannot in such a case imagine x to be a symbol of a variable number. Unless some intellect be in existence to mean something by x, or to make a symbol of x, there can be no such thing as a variable number, or as the abstract idea of number at all. When we say, let x be a variable number, we must always be understood to mean, let x be a symbol which at one time we may be allowed to make to stand for one number, and at another time for another.
Now as to gradual variation. A point never changes its distance from another by, say a foot, without making every assignable lesser change in the interval. Or, a line which is lengthened from A B to A c
by the motion of a point, must at some period of the change be equal to A D, if A D be anything between A a and A o. At least it is a neces sary condition of our existence to believe this to be as evident as that two straight lines cannot inclose a space, though [SrACE AND TIME] we believe some would be found to deny it. But in the case of number, we cannot form anything but an approximation to this idea of gradual variation. We can pass from 1 to 2 by successive steps, by millions of millions of steps if we please : that is, h representing a small fraction, we can proceed from 1 to 2 by the steps 1 +h, 1 + 2h, 1 +35, &c., in such manner that we shall not arrive at 2 till a million of million of steps have been made. But this is not gradual variation, such as is in our ideas when we think of a line increasing in length by the reces sion of one extremity from the other. Nor, if we subdivide our steps ever so far, can we, in counting, cease to make steps ; that is, we cannot imagine gradual variation of number. When, therefore, we talk of x standing for a number, which is also to represent the number of units in a variable length, we can only mean that our numerical progression can be made, if we please, by steps so small, that whatever length A D may represent, the linear representatives of some or other of the numerical steps by which we pass from the number in A n to the number in AC, may be made as near tOAD as we please. It is, no doubt, in this essential distinction between the ideas involved in the variation of number and in that of magnitude, that the existence of INCOMMENSURABLE quantities takes its rise.
The first steps of the Differential Calculus are often embarrassed by a mode of speaking which appears as if two different symbols were used for the same thing. " Thus," it is said, " let x be a variable, and y a function of that variable, such that p is always Then let x be changed into x+h, in consequence of which y becomes y + k ; so that y+k= (x+ hr." Now if x be the symbol of the variable quantity, which can only mean this, that both before the quantity has changed, and after, it is represented by x, how can it be allowed both to let x, as it were, imply its own variation in its very meaning and yet alter x into x+h to denote that x changes / The truth is that the language is incorrect; it should be as follows :--Let there be two variable quantities, one of which is always the square of the other let x be the value first given to one of the variables, and y to the other, so that Then let a new value x+h be given to the first variable, in consequence of which the second becomes y+k, so that In fact, x does not represent a variable quantity, hut a certain value given to a variable quantity.