BOOK II. § 5. Number. whatever may be the continuum which it is applied to measure. Acts of counting are the first condition of, or ingredient in, the establishment of units of measurement; and the establishment of an unit, or fixed standard always equal to itself, is in turn the first and indispensable step in measuring phenomena of any kind whatever. In this way calculation in its whole development is directly applicable to the divisions or boundaries of space, namely, by breaking them up into units of equal length, or portions measurable by one another. Hence figures in space can be expressed by numbers; units of length, breadth, and volume, being once taken. The measurement of time-lengths by means of constant and generally applicable units depends on the measurement of space-lengths as one of its conditions, and is therefore only indirectly possible.12 Hence Time, though (or rather because) it is the most fundamental condition of science generally, in its character as the one indispensable formal element of consciousness, escapes being the objectmatter of any special science of its own. For in order to its being conceived as an object at all, it must be taken along with some determinations or differences belonging to its inseparable content, or co-element, of feeling, since time strictly pure is an 12 See a paper by the late Edward Hawksley Rhodes, The Scientific Conception of the Measurement of Time, read before the Aristotelian Society, June 1, 1885, and published in MIND, Vol. X., p. 347, First Series. Perhaps I should also mention a paper by myself, Time-measurement in its bearing on Philosophy, published in the Proceedings of the Aristotelian Society, Vol. II., No. 2, Part 2, 1893.-I take this opportunity of gratefully acknowledging the assistance I have derived from conversations with my friend Mr. E. Hawksley Rhodes, during the latter years of his life, and also from correspondence with my friend (and quondam teacher in mathematic during his sojourn in England), M. Edouard Merlieux, on the subject of the present Section. Not that I mean in any way to cast upon them the responsibility for errors due to my own imperfect grasp of mathematical science, and still less for the course of my metaphysical speculations concerning it. abstraction, incapable of being even brought into consciousness without some reference to that from which it is abstracted by thought. And furthermore, in order to its being treated as the objectmatter of a special branch of science devoted to it alone, these determinations can only be drawn from that inseparable co-element of feeling which is exclusively its own, that is, from feelings which occupy time only, and not time and space together. But these feelings alone afford, as we have seen, no constant unit, applicable to the measurement of successive time-durations. Time differs in this respect from space, the inseparable content or co-element of which, I mean the element of visual and tactual sensations, is rich in differences of direction and magnitude, which can be brought into juxtaposition and measured one against another. And these spatial measurements are in fact the ultimate means which we possess for the measurement of time-durations, though this only indirectly. Ideally, indeed, we can imagine time divided into durations exactly equal to one another, and make of this an ideal standard, to which actual indirect measurements may be conceived to approximate. And this is in fact the very thing done by Newton, when he says of "absolute time," that it "flows equably," for that is equivalent to conceiving it divided into units of equal duration. In this conception of "absolute time," the science of time may be said at once to begin and end. As a body of doctrine it does not exist. Practically, however, its place is taken by the science of the Divisions of Time, that is, by Calculation, or the science of Pure Number. Pure BOOK II. CH. I. § 5. Number. BOOK II. § 5. Number. Calculation and Pure Geometry, founded respectively on the two formal elements in all consciousness, time-duration and space-extension, are the two sciences which stand at the fountain head of all positive and exact science. From the foregoing account of the applicability of abstract or pure Number to the measurement of particular contents or parts of that concrete timestream of consciousness, out of which, by attention and abstraction, it springs, we see not only the origin of the conception of Quantity in general, but also the origin of the two kinds or classes into which quantity is usually considered as divisible, namely, those of (1) continuous, (2) discrete quantity. Number is discrete quantity, in the sense of representing the result of ideally dividing continuous quantity into a plurality of parts, or smaller continua, though it must be remembered, that it is only by division that an originally undivided continuum becomes, or can be thought of as, a quantity at all. A number is the name for one or more of the parts arising from such divisions. Numbers being then considered as consisting of one or more unit numbers, and continuous quantities being reduced to measurement by dividing them into unit continua, the actual measurement of continuous quantity may be considered as answering the question How much, and the actual measurement of discrete quantity as answering the question How many? And the application of the latter to the former, when it can be effected, always tells us how many units of continuous quantity are to be found in the continuum which is measured. Continuous and discrete quantity are therefore, in strictness, not two separate classes of quantity, but two distinct though inseparable ways in which quantity may be regarded. Without continuity no quantity could possibly exist; without discreteness it could not be recognised as quantity. The very idea of quantity arises from the purposive introduction of an ideal division into a given representational continuum. § 6. The Infinity. 6. From these brief considerations on the mathematical sciences of Geometry, Kinematic, and Cal- Conception culation, I propose now to return to the relation which the phenomena of abstract space, time, and number, bear, when so treated, to the objectified panorama of the real external world, as presented to us by the metaphysical analysis of experience. And in the first place, the remarks now made on the subject of number enable us to state briefly how we come to attribute infinity to space and eternity to time, as they appear in that objectified panorama, and that in both directions, (1) divisibility, (2) extensibility, in infinitum. In the nature and origin of Number we have something wherewith to contrast, and by contrasting render intelligible, the infinity of time and space in both directions. Number is to time, as an objectified continuum, what geometrical figure is to space, as an objectified continuum; both are limitations introduced by thought into pre-supposed perceptual contents, which without them would be abstract continua; the perception of actual differences in the content, in both cases, being the circumstance which thought repeats in representation, and erects into an ideal limitation of space and time as abstract percepts. A series of numbers, say from 1 to 100, or on the VOL. II. F other side of zero from 1 to 100, corresponds to a closed geometrical figure, say a cube or a sphere. The process of counting, that is, naming by figures or symbols, any limited number of units in time alone, with abstraction from space, is what the process of imagining a closed solid geometrical figure is in space and time together. Now both these processes are processes of thought applied to perception, and both are conditioned as processes by the perceptions of time or of space, which are their pre-suppositions, and of which they are modifications. In number or in time, no number expressing a length of time can be named so great that you cannot add 1 to it, an ability which necessarily involves the existence of time beyond it; nor any fraction so small that you cannot decrease it by increasing its denominator, that is, suppose a shorter time within it. In space there is no closed figure so large that you can suppose there is no space beyond it, nor any so small that you cannot in thought interpose one still smaller between it and a mathematical point. Both processes may be carried on in both directions indefinitely, or in indefinitum; but they cannot be conceived as carried to completion, without contravening the conditions by which as processes they are constituted. In the case of the closed figure of space, to conceive the process of enlargement completed, by being carried to infinity, would be to conceive the figure annihilated as a figure. In a circle, for instance, the circumference would become in thought a straight line. In the case of numbering portions of time as continua, to conceive the process completed by being carried to infinity would be to destroy the |