end, particularly such an adjusting as requires comparison of different means to pick out the fittest, is the source of all quantitative ideas—ideas such as more and less, nearer and farther, heavier and lighter, etc. Quantity means the valuation of a thing with reference to some end; what is its worth, its effectiveness, compared with other possible means. These two conceptions-(a) the origin of quantitative ideas in the process of valuation (measuring) and (b) the dependence of valuation upon the adjusting of means to an end (i. e., ultimately upon activity) are the beginning of all conceptions of quantity and number, and the sound basis of all dealing with them.

THE IDEA OF BALANCE OR EQUATION.-We shall now note more definitely what is implied in the foregoing account the constant aiming at a balance or equivalence-(valens, worth; æquus, equal). The process of adjusting means to end is not simply a process of roughly estimating the value of certain things with regard to the end aimed at; but, as already said, it is economical and successful in the degree in which is employed just the amount of energy, just the amount of means necessary to accomplish the aim. The means used must just balance the end sought. Every machine, for example, represents an adjustment of certain means to a certain end. But there are good machines and bad machines. What constitutes the difference between a good machine and a poor one? The difference is found precisely in the fact that the former represents in itself and in the arrangement of its parts not only an adjustment in general to some end, but an accurate adjustment to the precise end to be reached; it is the embodiment

in wood and iron of a series of mechanical principles-it represents an equation. Now it is this necessity of exact balance or equivalency which transforms the vague quantitative ideas of smaller and greater, heavy and light, and so on, into the definite quantitative ideas of just so distant, just so long, so heavy, so elastic, etc. This demands the introduction of the idea of number. Number is the definite measurement, the definite valuation of a quantity falling within a given limit. Except as we count off means and end into just so many definite units, there can not be an economical adjustment, and there can not be a precise balance.

Summary.-Number arises in the process of the exact measurement of a given quantity with a view to instituting a balance, the need of this balance, or accurate adjustment of means to end, being some limitation.

Illustration.The logical steps of the development of number may be illustrated as follows: First, there is a recognition that one is distant from one's destination. —say, the camp. Next, that one can by travelling fast reach the camp by sunset. Third, the recognition that the present time (the time of starting) is such an hour of the day, e. g., two o'clock, and that sunset occurs at such a time-say seven o'clock; that the present spot is just so many miles distant from the camp; and that, consequently, one will have to travel just so many miles in a given time-say an hour-to reach the camp; this last stage being the equation or balance.

THE REASON FOR ABSTRACTION AND GENERALIZATION. We are now prepared to see the reason for the neglect of the sense qualities (the abstraction) and for the reference to the whole (the generalization) included

in all numbering. When we are regarding a thing not in itself, but simply as a means for some end, we take no account of any qualities which it may possess except this one quality of being related to the end. If I am to find out merely the quantity of land in a field, the fact that a part of the field is heavy clay and the rest rich, loamy soil is not taken into consideration; these qualities do not make the size value of the field, and are nothing to my purpose. I restrict attention entirely to the mathematical measurements, which in themselves are necessary and sufficient for the end to be reachedthe determination of the absolute area of the field. But if I am to compute the money value of the field, and know that the loamy soil has one value and the clayey soil another, these qualities, having a relation to the end in view, would have to be noted as controlling the measurements; while all other qualities-kind of clay, character of loam, moisture, and dryness-would be neglected as not bearing on the question of the money value of the field.

Similarly, if we want to know the whole amount of cloth in a store, we neglect all special qualities of cloth, and abstract the one quality of being cloth; silks, woollens, linens, cottons, however marked their differences, are alike in possessing this one quality that makes for our present purpose-they are all cloth. But if we are required to find the total money value, as in taking an inventory, and the different kinds of cloth have different prices, then we should abstract the special quality of being silk, or linen, or woollen, or cotton, and neglect all other qualities, colours, patterns, etc., which are believed to have no effect upon the price. In other words, it is

always the end in view which decides what qualities we shall pay attention to and what neglect. We abstract, or select, the special quality that helps with reference to this end. The rest, for purposes of measurement, are nothing to us.

It is obvious that it is the same reference to the end to be accomplished that constitutes generalization. We regard the various objects selected as having a relation, as making up one whole or class, because, no matter what their differences in themselves, they all serve the same end. It is this common service in helping towards one and the same end which binds them together, even if to eye or ear the things are entirely different. It is the reference to the end to be reached that controls both the abstraction and the generalization.

The books composing a library may be of many kinds-primers and dictionaries, novels and poems, printed in all languages, with pages of all sizes, and bindings in endless variety, yet as serving the one purpose of communicating intelligence through written or printed symbols they all fall together, and can be counted as making up one group.

THE PROCESS OF MEASURING.

We have now (1) noted the psychological processes of abstraction and generalization involved in all number, and have (2) traced them to the need of an economical adjustment of means to end which makes necessary the process of measuring from which number has its genesis. We have now to note in more detail the nature of this latter process.

Stages of Measurement.-We begin with the vague estimate of bulk, size, weight, etc., and go on to its accurate determination from the indefinite how much to the definite so much. It is the difference between saying that iron is heavy and that so much iron at a given temperature and a given latitude weighs just so much; or between saying that the blackboard is of moderate size and that it contains so many square feet. The development from the crude guess to the exact statement depends upon the selection and recognition of a unit, the repetition of which in space or time makes up and thus measures the whole. The savage may begin by saying that his camp is so many suns away. Here his unit is the distance he can travel between sunrise and sunset. He measures by a unit of action, but that unit is itself unmeasured—just how much it is he can not say or he measures by marking off so many paces. The pace is the unit, and is, relatively, more definite or accurate than the day's journey, but, absolutely, it is unmeasured. Only when the unit itself is accurately defined do we pass from vague quantity to precise numerical value.

Quantities in Different Scales of Measurement.But the process of measurement may be carried a step further. For accurate measurement the unit itself must be measured with a unit of the same kind of quantity; but the unit may also have a defined relation to a different kind of quantity. We are thus enabled to compare quantities lying in different scales of measurement. We can not, for example, directly compare weights and volumes, but we may compare them indirectly. If we discover, for instance, that four cubic inches of iron