Conceptions of Number not always definite.
There are however two practical instances, which may be cited to illustrate the position, that number is not a thing to be as matter of course definitely conceived in the mind. One of these is the case of very young children. To them the very lowest numbers are soon intelligible, but all beyond the lowest are not so, and only present a vague sense of multitude, that cannot be severed into its component parts. The distinctive mark of a clear arithmetical conception is, that the mind at one and the same time embraces the two ideas, first of the aggregate, secondly of each one of the units which make it up. This double operation of the brain becomes more arduous, as we ascend higher in the scale. I have heard a child, put to count beads or something of the sort, reckon them thus: ‘One, two, three, four, a hundred.’ The first words express his ideas, the last one his despair. Up to four, his mind could contain the joint ideas of unity and of severalty, but not beyond; so he then passed to an expression wholly general, and meant to express a sense like that of the word multitude.
But though the transition from number definitely conceived to number without bounds is like launching into a sea, yet the conception of multitude itself is in one sense susceptible of degree. We may have the idea of a limited, or of an unbounded, multitude. The essential distinction of the first is, that it might possibly be counted; the notion of the second is, that it is wholly beyond the power of numeration to overtake. Probably even the child, to whom the word ‘hundred’ expressed an indefinite idea, would have been faintly sensible of a difference in degree between ‘hundred’ and ‘million,’ and would have known that the latter expressed something larger than the former. The circumscribing outline of the idea apprehended is loose, but still there is such an outline. The clearness of the double conception is indeed effaced; the whole only, and not the whole together with each part, is contemplated by the mind; but still there is a certain clouded sense of a real difference in magnitude, as between one such whole and another.
And this leads me to the second of the two illustrations, to which reference has been made. That loss of definiteness in the conception of number, which the child in our day suffers before he has counted over his fingers, the grown man suffers also, though at a point commonly much higher in the scale. What point that may be, depends very much upon the particular habits and aptitudes of the individual. A student in a library of a thousand volumes, an officer before his regiment of a thousand men upon parade, may have a pretty clear idea of the units as well as of the totals; but when we come to a thousand times a thousand, or a thousand times a million, all view of the units, for most men, probably for every man, is lost: the million for the grown man is in a great degree like the hundred for the child. The numerical term has now become essentially a symbol; not only as every word is by its essence a symbol in reference to the idea it immediately denotes; but, in a further sense, it is a symbol of a symbol, for that idea which it denotes, is itself symbolical: it is a conventional representation of a certain vast number of units, far too great to be individually contemplated and apprehended. As we rise higher still from millions, say for example, into the class of billions, the vagueness increases. The million is now become a sort of new unit, and the relation of two millions to one million, is thus pretty clearly apprehended as being double; but this too becomes obscured as we mount, and even (for example) the relation of quantity between ten billions of wheat-corns, and an hundred billions of the same, is far less determinately conveyed to the mind, than the relation between ten wheat-corns and one. At this high level, the nouns of number approximate to the indefinite character of the class of algebraic symbols called known quantities.
In proportion as our conception of numbers is definite, the idea of them, instead of being suited for an address to the imagination, remains unsuited for poetic handling, and thrives within the sphere of the understanding only. But when we pass beyond the scale of determinate into that of practically indeterminate amounts, then the use of numbers becomes highly poetical. I would quote, as a very noble example of this use of number, a verse in the Revelations of St. John. ‘And I beheld, and I heard the voice of many angels round about the throne, and the beasts and the elders: and the number of them was ten thousand times ten thousand, and thousands of thousands[790].’ As a proof of the power of this fine passage, I would observe, that the descent from ten thousand times ten thousand to thousands of thousands, though it is in fact numerically very great, has none of the chilling effect of anticlimax, because these numbers are not arithmetically conceived, and the last member of the sentence is simply, so to speak, the trail of light which the former draws behind it.
Now we must keep clearly before our minds the idea, that this poetical and figurative use of number among the Greeks at least preceded what I may call its calculative use. We shall find in Homer nothing that can strictly be called calculation. He repeatedly gives us what may be termed the factors of a sum in multiplication; but he never even partially combines them, even as they are combined for example in Cowper’s ballad,
John Gilpin’s spouse said to her dear,
Though wedded we have been
These twice ten tedious years, yet we
No holiday have seen.