If for a moment number is understood in its most restricted sense as meaning positive integer number, the solution of a simple equation leads to an extension; ax - b = 0 gives x = (b/a), a positive fraction, and we can in this manner represent, not accurately, but as nearly as we please, any positive magnitude whatever; so an equation ax + b = 0 gives x = -(b/a), which (approximately as before) represents any negative magnitude. We thus arrive at the extended signification of number as a continuously varying positive or negative magnitude. Such numbers may be added or subtracted, multiplied or divided one by another, and the result is always a number. Now from a quadric equation we derive, in like manner, the notion of a complex or imaginary number such as is spoken of above. The equation x^2 + 1 = 0 is not (in the foregoing sense, number = real number) satisfied by any numerical value whatever of x; but we assume that there is a number which we call i, satisfying the equation i^2 + 1 = 0, and then taking a and b any real numbers, we form an expression such as a + bi, and use the expression number in this extended sense: any two such numbers may be added or subtracted, multiplied or divided one by the other, and the result is always a number. And if we consider first a quadric equation x^2 + px + q = 0 where p and q are real numbers, and next the like equation, where p and q are any numbers whatever, it can be shown that there exists for x a numerical value which satisfies the equation; or, in other words, it can be shown that the equation has a numerical root. The like theorem, in fact, holds good for an equation of any order whatever; but suppose for a moment that this was not the case; say that there was a cubic equation x^3 + px^2 + qx + r = 0, with numerical coefficients, not satisfied by any numerical value of x, we should have to establish a new imaginary j satisfying some such equation, and should then have to consider numbers of the form a + bj, or perhaps a + bj + cj^2 (a, b, c numbers [alpha] + [beta]i of the kind heretofore considered),--first we should be thrown back on the quadric equation x^2 + px + q = 0, p and q being now numbers of the last-mentioned extended form--_non constat_ that every such equation has a numerical root--and if not, we might be led to _other_ imaginaries k, l, &c., and so on _ad infinitum_ in inextricable confusion. But in fact a numerical equation of any order whatever has always a numerical root, and thus numbers (in the foregoing sense, number = quantity of the form [alpha] + [beta]i) form (_what real numbers do not_) a universe complete in itself, such that starting in it we are never led out of it. There may very well be, and perhaps are, numbers in a more general sense of the term (quaternions are not a case in point, as the ordinary laws of combination are not adhered to), but in order to have to do with such numbers (if any) we must start with them.