within a certain limit; the means are the partial activities, the units by which we realize this whole. To build a house of a certain kind and value, we must have just so many bricks, so many cubic feet of stone, so much lumber, so many days' work, etc. The house is the end, the goal to be reached; these things are the means. The house has been erected at a certain cost; the counting off and valuing of the units which enter into these different factors, is the only way to discover that cost. CHAPTER IV. THE ORIGIN OF NUMBER: SUMMARY AND APPLICATIONS. SUMMARY: Complete Activity and Subordinate Acts. -Through the foregoing illustrations-which are illustrations of one and the same principle regarded from different points of view-we are now prepared for the statement which sums up this preliminary examination of quantity. That which fixes the magnitude or quantity which, in any given case, needs to be measured is some activity or movement, internally continuous, but externally limited. That which measures this whole is some minor or partial activity into which the original continuous activity may be broken up (analysis), and which repeated a certain number of times gives the same result (synthesis) as the original continuous activity. This formula, embodying the idea that number is to be traced to measurement, and measurement back to adjustment of activity, is the key to the entire treatment of number as presented in these pages, and the reader should be sure he understands its meaning before going further. In order to test his comprehension of it he may ask himself such questions as these: The year is some unified activity—what activity does it represent ? At first sight simply the apparent return of the sun to the same point in the heavens-an external change; yet the only reason for attaching so much importance to this rather than to any other cyclical change, as to make it the unit of time measurement, is that the movement of the sun controls the cycle of human activities-from seedtime to seedtime, from harvest to harvest. This is illustrated historically in the fact that until men reached the agricultural stage, or else a condition of nomadic life in which their movements were controlled by the movement of the sun, they did not take the sun's movement as a measure of time. So, again, the day represents not simply an external change, a recurrent movement in Nature, but a rhythmic cycle of human action. Again, what activity is represented by the pound, by the bushel, by the foot? What is the connection between the decimal system and the ten fingers of the hands? What activity does the dollar stand for? If the dollar did not represent certain possible activities which it places at our control, would it be a measure of value? Why may a child value a bright penny higher than a dull dollar? And so on. Illustrations: Stages of Measurement. Suppose we wish to find the quantity of land in a certain field. The eye runs down the length and along the breadth of the field; there is the sense of a certain amount of movement. This activity, limited by the boundaries of the field, constitutes the original vague muchness the quantity to be measured-and therefore determines all succeeding processes. Then analysis comes in, the breaking up of this original continuous * The historical origin of these measures will throw light upon the psychological point. activity into a series of minor, discrete acts. The eye runs down the side of the field and fixes upon a point which appears to mark half the length; this process is repeated with each half and with each quarter, and thus the length is divided roughly into eight parts, each roughly estimated at twenty paces. The breadth of the field is treated in the same way. The eye moves along till it has measured, as nearly as we can judge, just as much space as equals one of the smallest divisions on the other side. The process is repeated, and we estimate that the breadth contains six of these divisions. Through these interrupted or discrete movements of the eye we are able to form a crude idea of the length and breadth of the field, and thus make a rough estimate of its area. The separate eye movements constitute the analysis which gives the unit of measurement, and the counting of these separate movements (units) is the synthesis giving the total numerical value. But the breaking up of the original continuous movement into minor units of activity is obviously crude and defective, and hence the resulting synthesis is imperfect and inadequate. The only thing we are certain of is the number of times the minor act has been performed; it is pure assumption that the minor act measures an equal length every time, and a mere guess that each of the lengths is twenty yards. In order, therefore, to make a closer estimate of the content of the field, we may mark off the length and breadth by pacing, and find that it is a hundred and seventy paces in length and a hundred and thirty paces in breadth. This is probably a more correct estimate, because (a) we can be much more certain that the various paces are practically equivalent to one another than that the eye movements are equal, and (b) since the pace is a more definite and controlled movement, we have a much clearer idea of how much the pace or unit of measurement really is. But it is still an assumption that the various paces are equal to one another. In other words, this unit of measure is not itself a constant and measured thing, and the required measurement is therefore still imperfect. Hence the substitution for the pace of some measuring unit, say the chain, which is itself defined; the chain is applied, laid down and taken up, a certain number of times to both the length and the breadth of the field. Now the minor act is uniform; it is controlled by the measuring instrument, and hence marks off exactly the same space every time.* The partial activity being defined, the resulting numerical value—say, eight chains by six chains-is equally definite. Besides, the chain itself may be measured off into a certain number of equal portions; we may apply a minor unit of measure—e. g., the link-until we have determined how many links make up the chain. By means of this analysis into still smaller acts, the meaning of the unit is brought more definitely home to consciousness.+ *Note how the two factors of space and time appear in all measurement, space representing concrete value, time the abstract number, and both, the measured magnitude. + If it be noted that all we have done here is to make the original activity of running the eye along length and breadth first continuously and then in an interrupted series of minor movements, more controlled and hence more precise, the meaning of the proposition (page 52) regarding the origin of measurement in the adjustment of |