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

But this mathematical measurement, this analysissynthesis, is still insufficient for complete adjustment of activity. What, after all, is the value of this measured quality? What is it good for? Until this question is answered there can not be perfect adjustment of activities. To answer this brings us to the third and final stage of number measurement. This field will produce, say, only so many bushels of corn at a given price per bushel; it is, therefore, not worth so much as a smaller field which will produce as much wheat at a larger price per bushel. Or, in addition to the mere size of the field, it may be necessary to take into account not only the value of the crop it will raise, but also the cost of tilling it. Here there must be a much more complete adjustment of activities. The analysis concerns not only so many square rods; it includes also the money value of the crop and the cost of its production. The synthesis compares the result of this complex measurement with the results of other possible distributions of energy. Analytically the conditions are completely defined; synthetically there can be a complete and economical adjustment of the conditions to secure the best possible results.

The measured quantity representing the unified (or continuous) activity is the whole or unity; the measuring parts, representing the minor or partial activities, are the components or units, which make up the unified whole. In all measurement each of these measuring parts in itself is a whole act-as a pace, a day's journey, etc. But in its function of measuring unit it is at once minor acts to constitute a comprehensive activity will be apparent

once more.

reduced to a mere means of constructing the more comprehensive act. The end or whole is one, and yet made up of many parts.

SUMMARY.-All numerical concepts and processes arise in the process of fitting together a number of minor acts in such a way as to constitute a complete and more comprehensive act.

1. This fitting together, or adjusting, or balancing, will be accurate and economical just in the degree in which the minor acts are the same in kind as the major. If, for example, one is going to build a stone wall, the use of the means—the minor activities—will not be accurate until one can find a common measure for both the means, the use of the particular stones, and the end, the wall. Size, or amount of space occupied, is this common element. Hence, to define the process in terms of just so many cubic feet required is economical; to describe it in terms of so many stones would be impossible unless one had first found the volumes of the stones. Hence, once more, the abstraction and the generalization involved in all numerical processes—the special qualities of the stone are neglected, and the only thing considered is the number of cubic feet in the stone-abstraction. But through this factor of so much size the stone is referred at once to its place in the whole wall and to the other stones-generalization.

2. An end, or whole of a certain quality, furnishes the limit within which the magnitude lies. Quantity is limited quality, and there is no quantity save where there is a certain qualitative whole or limitation.

3. Number arises through the use of means, or

minor units of activity, to construct an activity equal in value to the given magnitude. This process of constructing an equivalent value is numbering-evaluation. Hence, there are no numerical distinctions (psychologically) except in the process of measuring some qualitative whole.*

4. This measuring or valuation (defining the original vague qualitative whole) will transform the vague quantity into precise numerical value; it will accomplish this successfully in just the degree in which the minor activity or unity of construction is itself measured, or is also a numerical value. Unless it is itself a numerical quantity, a unity measured by being counted out into so many parts, the minor and the comprehensive activity can not be made precisely of the same kind. (Principle 1.)

5. Hence the purely correlative character of much and many, of measured whole and measuring part, of value and number, of unity and units, of end and

means.

EDUCATIONAL APPLICATIONS.

We have now to apply the principle concerning the psychological origin of quantity and number to education. We have seen (a) the need in life, the demand in actual experience of the race and the individual, which brings the numerical operations; the process of measuring, into existence. We have seen (b) what forms number is required to assume in order to meet the need, fulfil the demand. We have now to inquire how far

* The pedagogical consequences of neglecting this principle will be seen in discussing the Grube method, or use of the fixed unit.

these ideas and principles have a practical application in educational processes.

The school and its operations must be either a natural or an artificial thing. Every one will admit that if it is artificial, if it abandons or distorts the normal processes of gaining and using experience, it is false to its aim and inefficient in its method. The development of number in the schools should therefore follow the principle of its normal psychological development in life. If this normal origin and growth have been correctly described, we have a means for determining the true place of number as a means of education. It will require further development of the idea of number to show the educational principles corresponding to the growth of numerical concepts and operations in themselves, but we already have the principle for deciding how number is to be treated as regards other phases of experience.

THE TWO METHODS: THINGS; SYMBOLS.-The principle corresponding with the psychological law-the translation of the psychological theory into educational practice may be most clearly brought out by contrasting it with two methods of teaching, opposed to each other, and yet both at variance with normal psychological growth. These two methods consist, the one in teaching number merely as a set of symbols; the other in treating it as a direct property of objects. The former method, that of symbols, is illustrated in the old-fashioned ways-not yet quite obsolete — of teaching addition, subtraction, etc., as something to be done with "figures," and giving elaborate rules which might guide the doer to certain results called "answers."