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off eggs and oranges by the dozen, and weigh out flour by the pound. If everything that ministers to human. wants could be had by everybody just when wanted, we should never have to concern ourselves about quantity. If everything with which human activity is in any way concerned were unlimited, there would of course be no need to inquire respecting anything whatever: What are its limits? How much is there of it? Even if a thing were not actually unlimited, if there were always enough of it to be had with little or no expenditure of energy, it would be practically unlimited, and hence would never be measured. It is because we have to put forth effort, because we have to take trouble to get things, that they are limited for us, and that it becomes worth while to determine their limits, to find out the quantity of anything with which human energy has to do.

Limit, in other words, is the primary idea in all quantity; and the idea of limit arises because of some resistance met in the exercise of our activity.

Economy of Energy.-Because we have to put forth effort, because we are confronted by obstacles, our energy is limited. It therefore becomes necessary to economize our energy-that is to say, to dispose of it or distribute it in such ways as will accomplish the best possible results. This economy does not mean a hoarding up or withholding of energy, but rather giving it out in the most effective way, husbanding "our means so well they shall go far." If we put forth more energy than is needed to effect a certain purpose, and equally if we put forth less than is needed, there is waste; we fail to make the most of the resources at our disposal. We carry out our plans most successfully, and perform the

hardest tasks with the least waste of power when we accurately adjust our energies to the thing required. Because of the limitation of human energy all activity is a balancing of energy over against the thing to be done, and is most fruitful of results when the balancing is most accurate. If the arrow of the savage is too heavy for his bow, or if it is too light to pierce the skin of the deer, there is in both cases a waste of energy. If the bow is so thick and clumsy that all his strength is required to bend it, or so slight or uneven that too little momentum is given to the arrow, there is but a barren show of action, and the savage has his labour for his pains. Bow and arrow must be accurately adjusted to each other in size, form, and weight; and both have to be equated (as the mathematician would say) or balanced to the end in view-the killing of the game. This involves the process of measurement, and its result is more or less definite numerical values.

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Means and End: Valuation.-The same principle be otherwise stated in terms of the relation existing between means and end. If all our aims were reached at the moment of forming them, without any delay, postponement, or countervening occurrences-if to realize an end we had only to conceive it—the necessity for measurement would not exist, and there would be no such thing as number in the strictly mathematical sense. But the check to our activity, the limitation of energy, defers the satisfaction of our needs. The end to be realized is remote and complex, and in using adequate means, distance in space, remoteness in time, quantity of some sort has to be taken into account, and this means accurate measurement.

In working out a certain purpose, for example, one of a series of means is a journey to be undertaken; it is of a certain length; it is to be completed in a given time, and within a certain maximum of expense, etc.; and this involves careful calculation-measurement and numerical ideas. In brief, it may be said that quantity enters into all the activities of life, that the limitation. of energy demands its economical use-that is, the precise adjustment of means to end—and that such use, such careful adjustment of activity, depends upon exact measurement of quantity.

The child and the savage have very imperfect ideas of number, because they are taken up with the things of the present moment. There is no imperative demand for the economical adjustment of means to end; living only in and for the present, they have no plans and no distant end requiring such an adjustment. They do not ask how the present is to be made of use in attaining some future or permanent good. But as soon as the child or the savage has to arrange his acts in a certain order, to prescribe for himself a certain course of conduct so as to accomplish something remote, then the idea of quantity begins to exist. When a savage is aimlessly playing with a stick, he does not connect it with any desired end, and accordingly does not reflect upon its quantitative value. But if he wants to shape it into an arrow, then the measuring (the quantifying) immediately begins; he examines several sticks; he thinks this one too big, that too small; this too brittle, that too elastic; this one of the right size, weight, and elasticity -all of which are simple quantitative ideas.

Thus it is also in the case of a child playing with

stones; so long as he is contented with them just as they are, not thinking of them as means towards a definite end, ideas as to their weight, or size, or number, do not enter into his mind. But if he decides to throw at a mark, a rude measurement or valuation begins; this is too heavy or too large, that too light or too small, etc. Or if he wishes to build a house, or to form an inclosure with them, he begins at once to note size and shape, and perhaps to form a vague notion of the number required for his ideal house or inclosure. Having only the vaguest ideas of quantity and number, he can not accurately compare means with end, and his first efforts at building will be purely tentative. His ideal "playhouse" or sheepfold, or garden, is too large for his means; he has not stones enough to complete the house or to “go round" the inclosure; he must build on a smaller scale. Or the structures are completed without exhausting the materials, and with the remaining stones he puts together another piece of work, perhaps an addition to house or garden. In all this there is more careful comparison of quantities, a better adjustment of means to end, closer measurement, and some approach to definite numerical ideas.

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Again, the savage has to take a journey to find his game. The remoteness of the hunting ground makes him a measurer"; he must think how distant the hunting ground is, how long he will be gone, how much he should carry with him, etc. If he has a choice of ways of reaching his hunting ground, the numerical valuation becomes more marked; short and long, near and far become more closely defined. The comparison of different means as to their serviceable

ness in reaching an end not only gives us a vague idea of their quantity, but tends to make it precise, numerical.

The importance of this process of comparing different means in order to select the best, in the development of number judgments, may be illustrated as follows: A chick just out of the shell will peck accurately at a grain of corn or at a fly. We might say that it measured the distance. But this does not mean that it has any idea of distance or that there is any conscious process of estimating its extent. There is, in reality, no measuring, no comparison, no selection, but simply direct response to the stimulus; and there is, therefore, no sense of distance. So an average child by the time he is six months old will reach out only for objects which fall within the length of his arms, while previously he may have attempted to grasp objects irrespective of their distance. Yet he does not measure, or necessarily have a consciousness of distance, unless there happen to be, say, two objects, one just within his reach, the other just beyond, and he selects the nearest object as a result of comparing the distances. If the child does this, he performs a rudimentary measurement and has a crude idea of distance. So, too, creeping, walking, etc., imply what may be termed measurement, but they involve no process of measuring, and hence no consciousness of space values, of length, or size, or form, until the child begins to prefer and select one of several paths. When he does this, he refers the various paths to his own ease of action, and thus gets a standard for comparison.

Summary. The conscious adjusting of means to

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