weigh as much as twenty-nine cubic inches of water, four cubic inches of gold as much as seventy-seven cubic inches of water, and two cubic inches of honey as much as three cubic inches of water, we have then the means of comparing cubic inches of iron with pennyweights of gold or with pounds of honey. Thus the different scales of volume and weight are brought together by comparing both with a common standard. Take a case of comparing money values. We can directly compare the cost of three yards of calico with that of nine yards of the same quality; but we can not directly compare-as to cost-lengths of calico of different qualities, or a length of calico with one of silk. But if we know that one yard of calico is worth (is) measured by) eight cents, and one yard of silk worth one dollar and sixty cents, we can accurately measure the worth of any quantity of calico in terms of any quantity of silk. If it were not for this discovery of a unit differing in kind from the quantity to be measured, and yet capable of comparison with it, our exact measurements would always be confined within one and the same scale-time, weight, volume, etc.; we should simply have to guess how much of one scale would equal a given quantity of another. Moreover, the measurement within a given scale is imperfect if we have no means of defining some unit of the scale in terms of a different quantity. We may know how much a pound is in terms of an ounce, an ounce in terms of drachms, but we can never get out of this circle. We can never know how much the ounce, the pound, etc., really is; our measurement can not reach the highest stage of development-what may be called the scientific stage. There is perfect measurement only when this stage is reached. The pound of the weight scale, for example, is not perfectly defined in terms of weight (ounces, etc.) alone; the pound is more accurately defined when we discover that it is, say, the amount of copper which under certain conditions will displace such and such an amount of water. Only in some such way as this is our unit ultimately defined, and only when the unit of measure is itself perfectly measured can there be perfectly exact or scientific measurement. This measurement of a quantity in terms of quantity unlike in kind, but alike in some one respect,* is the completion of number as the tool of measurement. Beyond this stage, number can not go, but until it has developed to this point it is an imperfect instrument of measurement. There are therefore three stages of measurement: 1. Measuring with an undefined unit, as in measuring length by the unit "pace," apples by the unit apple, etc. 2. Measuring with a unit itself defined by comparison with a unit of same kind of quantity—the yard, the pound, the dollar, etc. 3. Measuring with a unit having a definite relation to a quantity of a different kind. Counting and Measuring.-It has been said that number originates from measurement; that it is a statement of the numerical value of something. But we are accustomed to distinguish counting (i. e., numeration, * This common basis of comparison is always, ultimately, movement in space. numbering) from measuring. It is usually said that we count objects, particular things or qualities, to see how many of them there are, while we measure a particular object or quality to see how much of it there is. We count chairs, beds, splints, feet, eyes, children, stamens, etc., simply to get their sum total, the how many; we measure distance, weight, bulk, price, cost, etc., to see how much there is. Some writers say that these "two kinds" of quantity, which they call quantity of magnitude (how much) and quantity of multitude (how many), are entirely distinct. Nevertheless, all counting is measuring, and all measuring is counting. When we count up the number of particular books in a library, we measure the library-find out how much it amounts to as a library; when we count the days of the year, we measure the time value of the year; when we count the children in a class, we measure the class as a whole-it is a large or a small class, etc. When we count stamens or pistils, we measure the flower. In short, when we COUNT We measure. On the other hand, in measuring a continuous quantity—“quantity of magnitude"--counting is equally necessary. We may apply a unit of measure to such a quantity and mark off the parts with perfect accuracy, but there is no measurement till we have counted the parts. Thus, the only way to measure weight is by counting so many units of density; distance, by counting so many particular units of length; cost or price, by counting so many units of value-dollars or what not. In other words, when we MEASURE we count. The difference is that in what is ordinarily termed counting, as distinct from measuring, we work with an undefined unit; it is vague measurement, because our unit is unmeasured. When we say ten apples, five books, six horses, etc., we measure some whole, some how much, by counting its parts, the how many; but we do not know just how much one of these parts or units is. If we knew the exact size, or weight, or price of the books or apples, we should have a more accurate measurement and a more accurate valuation.* On the other hand, what we ordinarily call measuring, as distinct from counting, is simply counting with a unit which is itself measured by so many definite parts. If I count off four books, "book," the unit which serves as unit of measurement, is itself only a qualitative, not a quantitative unity, and the quantity four books is not a definitely measured quantity. If I say each book weighs six ounces or is worth sixty cents, the unit of measurement is itself both qualitative and quantitative; and the price or the weight of the four books is a definitely measured quantity. We shall see hereafter that, strictly speaking, merely qualitative wholes used as units give only addition and subtraction; that the whole which is itself quantitative, as well as qualitative, gives multiplication and division. If, however, the wholes are taken or assumed as equal in value, then, of course, the operations of multiplication and division may be performed with them. But this is only because the assumption of equal (measured) * It is a great pity that our authorities use these unmeasured units so much, particularly in fractions. Half an apple, half a pie, is a practical, not a mathematical expression at all. To make it mathematical we should have to know just how great the whole ishow many ounces or cubic inches. value in the units is made. If we are to divide fifteen apples "equally" among five boys, giving each boy three apples, this "equal" distribution assumes the equality of the units (apples) of measure. Much and Many.-The whole falling within a certain limit supplies the muchness; for example, the amount of money in a purse, the amount of land in a field, the amount of pressure it takes to move an obstacle, etc. This "much," or amount, is vague and undefined till measured; it is measured by counting it off into so many units. We "lay off" distance into so many yards, and then we know it to be so much. We reckon up the pieces of money in the purse and know how much their value is. A man has a pile of lumber; how "much" has he? If the boards are of uniform size, he finds the number (how many) of feet in one board, and counts the number (how many) of boards, and finds the whole so many feet-that is, the indefinite "how much" has become, through counting, the definite "so much." Then, again, if he wishes to find the money value of the lumber, how much it is worth, he must count off the total number of feet at so much (so many dollars) per thousand, and the resulting so many dollars represents the worth of the lumber. The many, the counting up of the particular units, measures the worth of the whole. The counting has no other meaning, and the measurement of value can occur in no other way. It is clear that these two sides of all number are relative to each other, just as means and ends are relative. The so many measures the so much, just as the means balance the end. The end is the whole, all that comes |