full statement of the process of measurement; it means (1) that the primary unit of measurement (the standard of reference) is one yard, (2) that the derived unit of measurement is one third of this, and (3) that this derived unit is taken nineteen times to measure the quantity. This is seen to agree with the mental process of the exact stage of measurement in which the unit of measure is itself defined or measured (see page 94). There must be, as we have seen, (1) a standard unity of reference (the primary unit), (2) a derived' unit (the unit of direct measurement), and (3) the number of these in the quantity. The fraction gives complete expression to this process: In $1, for example, (1) the dollar is the unit of reference; (2) it is divided into four parts to get the derived unit-the actual unit of measure; (3) the "numerator" 3 shows how many of these units make up the given quantity, and expresses the ratio of this quantity to the standard unity.

So, again, the measurement-19 feet-of the side of a room can be stated in terms of other units of the scale. It is 12 x 19 inches, or 19 ÷ 3 yards, and the first of these expressions, as well as the second, is one of fractions; it is 228—that is, not 228 ones merely, but 228 of a definite unit of measure—namely, one twelfth of a foot; just as the second is 12-i. e., 19 times a unit of measure defined by its relation to the yard. In the former case we do not generally state the measurement in fractional form, but the interpretation of it demands an explicit reference to a denominator. Note what this brings us to: 19 (feet) 228 (inches) =(yards) =202 (rods)—that is, four entirely different numbers equal to one another; a result which must appear

=

utterly meaningless to a child who has been trained by the fixed unit method. Any method which treats number as a name for physical objects can not but reach just such absurdities. Only the method which recognises that number is a psychical process of valuation (analysis-synthesis) is free from such difficulties. The unit does not designate a fixed thing; it designates simply the unit of valuation, the how much of anything which is taken as one in measuring the value (or how much) of a group or unity. It defines how many units each of so much value make up the so much of the whole. The complete process is one of fractions, and the full statement of it is a fraction, whether written out in full or necessarily understood in the interpretation. The 228 inches is 228, signifying that the number of the derived units of measure in one inch is 1; 19 feet is yards, signifying that the number of the derived units of measurement in one yard is 3; the 209 rods show that the number of units of measurement in one rod is 11; in other words, the unit of measure in 12 is one of the three equal parts of one yard, etc.

It appears, therefore, that every numerical operation which makes a vague quantity definite, when fully stated, involves the "terms" of a fraction—that is, a fraction may be considered as a convenient language (notation) for expressing quantity in terms of the process which measures or defines it-which makes it "number."

A fraction, then, completely defines the unity of reference, and thus determines the unit of measure for the quantity that is to be measured. Thus the inch may be defined from foot, the foot from yard, the ounce from + pound, the cent from 188 dollar, etc. In each

case the denominator shows the analysis of a standard unity into units of measurement-i. e., the unity in terms of the units taken collectively. Thus the measurements of the quantities 7 inches, 5 ounces, 35 cents are more explicitly stated by the respective fractional forms foot, pound, dollar, because the unit of measure in each case is consciously defined by its relation to a standard unity in the same scale.

It is clear that the definition of number (page 71) includes the fraction, for in both fraction and integer the fundamental conception is that of a quantity measured by a number of defined parts-the conception of the ratio of the quantity to the measuring unit. The fraction differs from the integral number-in so far as it differs at all-in defining the measuring unit, and thus giving more completely the psychical operation in the exact stage of measurement.

If the fraction, as being a number, is a mode of measurement, there appears to be no need of a special definition of it as the foundation of a new or different class of numerical operations. The definitions which ignore fractions as a mode of measurement are in general vague and inaccurate, and lead to much perplexity in the treatment of fractions. It is hardly accurate to say that a "fraction is a number of the equal parts of a unit," or that "it originates in the division of the unit into equal parts." Here the important distinction between unity and a unit is overlooked. Measuring a piece of cloth we find it contains four yards: before measurement it was mere unity, after measurement a defined unity; but in neither case is it a unit. It is, after measurement, a unity of units-a sum. Nor is it entirely

consistent with the measuring idea to say that a fraction is one or more of the equal parts of a unity. Of course, in counting the equal parts of a measured whole —a unity—we take a number of parts in making the synthesis of all the parts. But since a fraction is a number, and therefore denotes measured quantity, it denotes a whole quantity, a unity—e. g., of a yard is as much a quantity-a measured unity-as 4 yards or 40 yards; it is a fraction in its relation to a larger unity, the yard taken as a standard of reference.

The Improper Fraction.-From the same misapprehension of the nature of number endless discussions arise regarding the classes of fractions "proper,” “improper," etc. With a right conception of the measuring function of a fraction there is no mystery about the improper improper" fraction. From the definition of a fraction as a "number of the equal parts of a unit," it is inferred, e. g., that of a yard can not be a fraction, because it represents not parts of a unit, but the whole unit and something more. Since 3 thirds make up the yard (the unit), whence come the 4 thirds?

In this objection we have the fallacy of the fixed unit as well as the misapprehension of the nature of number. The fraction in the expression of a yard is a number. It means the repetition' of a unit of measure to equal a certain quantity. This unit of measure is not the yard; it is a unit defined by its relation to the yard; it is one of the three equal parts into which the unity yard is divided to get the direct unit of measure; and there is absolutely nothing to make the yard the limit of quantity to which this unit can be applied. The yard is the primary unit of reference from which

the actual measuring unit is derived, and there is no more mystery in the application of this unit to measure a quantity greater than the primary unit than in the measured quantity, 3 feet × 4, because it is greater than one foot, the primary unit from which the measuring unit (3 feet) is derived.

The expression 4 thirds of a yard indicates an exactly measured quantity; exactly measured, because the unit of measure is itself measured in its relation to another quantity of the same scale. This properly defined unit (1 third of a yard) can be applied to any homogeneous quantity whatever, and may be contained in such quantity one, two, three, four, n times; in fact, 4 thirds yard, 5 thirds yard, . n thirds yards are only different and more exact ways of stating the measurements-4 feet, 5 feet,

...

n feet.

The Compound Fraction.-Nor is there any dif ficulty in interpreting a "compound" fraction. The value of 8 yards of cloth at $ a yard is expressed by $× 8, a measurement which ought to occasion no more perplexity than $3 x 8, when it is understood that the denominator merely defines the unit of measure with reference to the primary unit. So the value of yard of cloth at $8 a yard is expressed by $8 × §, a measured quantity where, once more, the denominator shows how the unit of measure is to be obtained—i. e., it shows which of the myriad ways of parting and wholing $8-the unity of reference-will give the direct or absolute unit of measure. This explanation applies to $×, and to any compound fraction whatever.

The Complex Fraction.—It is said that the complex fraction is an impossibility, because a quantity can not