as far as five, e. g., he is conscious that five is connected with what goes before. This perception is one of moreness or lessness, of aggregation; five is more than four, and at last, definitely, it is one more than four. With the continuance of the physical acts there is further growth towards the higher conception. He separates a whole into parts and remakes the whole; he combines (using intuitions) unequal groups of measuring units (e. g., 3 feet and 4 feet) to express them as ones; he counts by ones, groups of two things, of three things, etc., and at last the idea of times, of pure number, is definitely grasped. The "five" is no longer merely one more than four, it is five times one, whatever that one may be. In other words, he has passed from the lower idea to the higher; from the idea of mere aggregation to that of times of repetition; from addition to multiplication.

It is plain that there must be time for the development of this abstracting and generalizing power. In fact, the complete development of the "times" idea, this factor relation, corresponds with the stages of the measuring power of number. The higher power of numerical abstraction is the higher power of the tool of measurement. This normal growth in the power of abstracting and relating can not be forced by any—the most minute and ingenious-analyses on the part of the teacher. The learner may indeed be drilled in such analyses, and may glibly repeat as well as "reason out" the processes; just as he can be drilled to the repetition of the words of an unknown tongue, or any other product of mere sensuous association. But it does not follow that he knows number, that he has grasped the

idea of times. The difficulty is not in the word times, as some appear to think; it is in the idea itself, and would not disappear even if the word were (as some propose) exorcised from our arithmetics. It has not yet been proposed to eliminate the idea itself—i. e., the idea of number-from the science of number.

SUMMARY. (1) Counting is fundamental in the development of numerical ideas; as an act or operation with objects it is at first largely a mechanical process, but with the increase of the child's power of abstraction it gradually becomes a rational process. (2) From this (partly) physical or mechanical stage there is evolved the relation of more or less, and addition and subtraction arise—that is, e. g., five is one more than four. (3) The addition, through intuitions, of unequal (measured) quantities, which are thus conceived and expressed as a defined unity of so many ones, is an aid to the development of the times idea. (4) Continuance of such operations-appealing to both eye and ear-brings out this idea more definitely-e. g., five is not now simply one more than four, it is five times one. (5) Counting (by ones) groups of twos, threes, etc., brings out still more clearly the idea of times. (6) Through repeated intuitions, sums (the results of uniting equal addends) become associated with times, the factor idea (times of repetition) displaces the part idea (aggregation), and multiplication as distinct from addition arises explicitly in consciousness.

The Process of Multiplication. The expression of measured quantity has, we have seen, two components, one denoting the unit of measure, and the other de

noting the number of these units constituting the quantity. But since the unit of measure is itself composed of a definite number of parts-is definitely measured by some other unit-it is clear that we actually conceive of the quantity as made up of so many given units (direct units of measure), each measured by so many minor units. For convenience we may call these minor units "primary," as making up the direct unit of measure, and this direct unit, as being made up of primary units, may be called the "derived" unit. We shall thus have in the complete expression of any measured quantity, (1) the derived unit of measure, (2) the number of such units, and (3) the number of primary units in the derived unit of measure.

For example, take the following expressions of quantity: In a certain sum of money there are seven counts of five dollars each; here the derived unit of measure is five dollars, the number of them is seven, and the primary unit is one dollar. The cost of a farm of sixty acres at fifty dollars an acre is sixty fifties; here the derived unit of measure is fifty dollars, the number of them sixty, and the primary unit one dollar. The length of a field is fifteen chains-that is (in yards), fifteen twenty-twos; here the derived unit is twenty-two yards, the number of them fifteen, and the primary unit one yard. In the quantity $x the primary unit is $1, the derived unit $4, and is the number expressing the quantity in terms of the derived unit.

Now, when a quantity is expressed in terms of the derived unit, it is often necessary or convenient to express it in terms of a primary unit. Thus, in the foregoing examples, the sum of money expressed as seven

fives may be expressed as thirty-five ones; the cost of the farm, expressed as sixty fifties, may be expressed as three thousand ones; and the length of the field, expressed as fifteen twenty-twos, may be expressed as three hundred and thirty ones (yards); and $× is measured by in terms of the primary unit. In each of these cases the second expression of the measured quantity merely states explicitly the number of minor (or primary) units which is implied in the first expression. The operation by which we find the number of primary units in a quantity expressed by a given number of derived units is Multiplication. It is plain that the idea of times (pure number, ratio) is prominent in this operation; we have the times the primary unit is taken to make up the derived unit, and the times the derived unit is taken to make up the quantity. The multiplicand always represents a number of (primary) units of quantity; the multiplier is always pure number, representing simply the times of repetition of the derived unit. But from the nature of the measuring process the two factors of the product may be interchanged, the times of repetition of the primary unit may be commuted with the times of repetition of the derived unit; in other words, the number which is applied to the primary unit may be commuted with the number which is applied to the derived unit.

Correlation of Factors.-In our conception of measured quantity these two ideas are, as has been shown, absolutely correlative. Measuring a line of twelve units by a line of two units, the numerical value is six; if we consciously attend to the process, the related conception instantly arises; we can not think six times two units

without thinking two times six units, because we can not think one unit six times without thinking one whole of six units. So, in measuring a rectangle 8 inches long by 10 inches wide, we can not analytically attend to the process which gives the result of 8 square inches taken ten times without being conscious of the inevitable correlate, 10 square inches taken eight times. In general: To think the measurement of any quantity as b units taken a times, is to think its correlate a units taken 6 times; for b units is b times one unit, and every one in b is repeated a times, giving a units once, a units twice, etc.-that is, a units b times.

EDUCATIONAL APPLICATIONS.

1. Just as, in addition, we must always begin with a vague sense of some aggregate, and then go on to make that definite by putting together the constituent units, while in subtraction we begin with a defined aggregate and a given part of it, and go on to determine the other parts; so, in multiplication, we begin with a comparatively vague sense of some whole which is to be more exactly determined by the "product," while in division we begin with an exactly measured whole, and go on to determine exactly its measuring parts. In multiplication the order is as follows: (1) The vague or imperfectly defined magnitude; (2) the definite unit of value (primary unit), which has to be repeated to make the derived (direct) unit of measure-the multiplicand; (3) the number of times this derived unit is to be repeated -the multiplier; and (4) the product-the vague magnitude now definitely measured.

The operation of multiplication, therefore, already