whether it is times or measuring parts he is to search for, before he begins the operation; to this knowledge the different names afford him no aid whatever.*

Partition, like Division, depends on Subtractions. -It is said, indeed, that in "partition" we are searching for the numerical value of one of a given number of equal parts which measure a quantity, and as a number can not be subtracted from a measured quantity, the problem can not be solved by division. To this the answer is easy: In the first place, the divisor in the arithmetical operation can be a number, and the subtractions rationally explained (see page 122). And, besides, we can by the law of commutation concrete the number, find the related factor, and properly interpret the result. But, in the second place, if the divisor can not be an abstract number, what magic is there in a strange name to bring the impossible within the easy reach of childhood? It seems, according to the partitionists, that 20 feet 5 feet represents a possible and intelligible operation; but that 20 feet 5 becomes possible and intelligible only by calling the implied operation a case of "partition"; it is then simply one fifth of 20 feet-that is, 4 feet. Certainly, if we know the multiplication table, we know that one fifth of 20 feet is 4 feet, but we know equally well that 5 feet is one fourth of 20 feet. These are not typical cases for the argu

* Owing to the fixed unit fallacy, the theory of the "two divisions" makes an unwarranted distinction between the actually measuring part and its times of repetition. The measuring part, as well as the whole, involves both the spatial element (unit of quantity) and the abstract (time) element; it is itself a quantity that is measáred by a minor unit taken a number of times.

ment; though attention to the processes even in these cases (see page 122) will show that if 20 feet ÷ 5 is impossible because "division" is a process of subtraction, so also is the process one fifth of 20 feet, because "partition" is equally a process of subtraction (page 122). For example, the operation indicated in $14899623 ÷ $4681 it is admitted involves subtraction-i. e., the separation of the dividend into parts, and the obtaining of partial quotients. But it is clear that 1-4681th of this dividend (partition) is obtained by exactly the same process—i. e., in both cases we have a first subtraction of 3000 times the divisor, a second of 100 times, a third of 80 times, and a fourth of 3 times, getting the same numerical quotient of 3183 in both operations; but 3183 is interpreted as pure number in the first case, and as measured guantity in the second-the so-called partition.

In fine, when it comes to pass that there can be a clear conception of a foot as measured by inches without the thought of both the factors, one inch and twelve times, then, but not till then, it may be rationally affirmed that the "two divisions" are radically different and totally unrelated processes.

FRACTIONS.

The process of fractions as distinguished from that of "integers" simply makes explicit—especially in its notation—both the fundamental processes, division and multiplication (analysis-synthesis), which are involved in all number.

In the fundamental psychical process which constitutes number, a vague whole of quantity is made definite by dividing it into parts and counting the

parts. This is essentially the process of fractions. The "fraction," therefore, involves no new idea; it helps to bring more clearly into consciousness the nature of the measuring process, and to express it in more definite form. The idea of ratio-the essence of number-is implied in simple counting; it is more definitely used in multiplication and division, and still more completely present in fractions, which use both these operations. Fractions are not to be regarded as something different from number-or as at least a different kind of number —arising from a different psychical process; they are, in fact, as just said, the more complete development of the ideas implied in all stages of measurement. So far as the psychical origin of number is concerned, it would be more correct to say that "integers" come from fractions than that fractions come from integers. Without the "breaking" into parts and the "counting" of the parts there is no definitely measured whole, and no exact numerical ideas; the definite measurement is simply (a) the number of the parts taken distributively (the analysis), and (b) the number of them taken col lectively (the synthesis). The process of forming the integer, or whole, is a process of taking a part so many times to get a complete idea of the quantity to be measured; and at any given stage of this operation what is reached is both an integer and a fraction—an integer in reference to the units counted, a fraction in reference to the measured unity.

Even in the imperfect measurement of counting with an unmeasured unit, the ideas of multiplication and division (and therefore of ratio and fractions) are implied in the operation. We measure a whole of

fifteen apples by threes; we count the parts-i. e., relate or order them to one another, and to the whole from which and within which we are working. This counting has a double reference-i. e., to the unit of measure, and to the whole which all the units make up. When, for example, we have counted two, three, we have taken one unit of measure two, three, times, and each count is expressed or measured by the numbers two, three, . . . —i. e., by "integers"; but also in reaching any of these counts we have-in reference to the whole-taken one of the five, two of the five, three of the five, etc.; that is, one fifth of the whole, two fifths, three fifths, etc.

No Measurement without Fractions.-When we pass to measurement with exact units of measure, this idea of fractions-of equal parts making up a given whole becomes more clearly the object of attention. The conception, 3 apples out of 5 apples (three fifths of the whole) has not the same degree of clearness and exactness as that of 3 inches out of a measured whole of 5 inches. Why? Because in the former case we do not know the exact value, the how much of the measuring unit; in the latter case the unit is exactly defined in terms of other unities larger or smaller; in 3 apples the units are alike; in 3 inches the units are equal. So in measuring a length of 12 feet we may divide it into 2 parts, or 3 parts, or 4 parts, or 6 parts, or 12 parts; then we can not really think of the 6 parts as making the whole without thinking that 1 is one sixth; 2, two sixths; 3, three sixths, etc. In the process of inexact measurement the idea of fractions is involved in that of exact measurement,

:

cess.

this idea is more clearly defined in consciousness. In short, wherever there is exact measurement there is the conception of fractions, because there is the exact idea of number as the instrument of measurement. The process of fractions, as already suggested, simply makes more definite the idea of number, and the notation employed gives a more complete statement of the analysissynthesis, by which number is constituted. The number 7, for example, denotes a possible measurement; the number states more definitely the actual proIt not only gives the absolute number of units of measure, but also points to the definition of the unit of measure itself—that is, the 7 shows the absolute number of units in the quantity, while the 10 shows a relation of the unit of measure to some other standard quantity, a primary unit of reference by which the actual measuring unit is defined. If a quantity is di-• vided into 2 equal parts, or 3 equal parts, or 4 equal parts, or n equal parts, the 2, 3, 4, . . . n shows the entire number of parts in each measurement, and corresponds with the "denominator" of the fraction which expresses the measured quantity as unity; and in counting up (e-numerating) the parts (units) we are constantly making "numerators "-e. g., 1 out of n, 2 out of n, 3 out of n, etc.; or 1-nth, 2-nths,

n

n-nths, or

n'

which is the measured unity. Or, if attention is given to the measuring units-the ones the parts are expressed by 1, 2, 3, etc., and the measured quantity itself

n

is expressed by Again, measuring the side of a

certain room, we find it to contain 2 yards. This is a