the mind constructs defined parts into a unified and definite whole. Absolute value (quantity numerically defined) is represented by the application of this how many to magnitude, to quantity-that is, to limited quality. To take an example of the confusion referred to: we are told that division is dividing a (1) number into a (2) number of equal (3) numbers. This definition as it stands has absolutely no meaning; there is confusion of number with measured quantity. Doubtless the definition is intended to mean: division is dividing a certain definite quantity into a number of definite quantities equal to one another. Only in (2), in the definition as quoted, is the term number correctly used; in both (1) and (3) it means a measured magnitude. A measured or numbered quantity may be divided into a number of parts, or taken a number of times; but no number can be multiplied or divided into parts. Number simply as number always signifies how many times one 'so much," the unit of measurement, is taken to make up another "so much," the magnitude to be measured. It is, as already said, due to the fundamental activities of mind, discrimination, and relation, working upon a qualitative whole; and we might as well talk of multiplying hardness and redness, or of dividing them into hard and red things, as to talk of multiplying a number or of dividing it into parts.

It may be observed that the problems constantly used in our arithmetics, multiply 2 by 4, divide 8 by 4, are legitimate enough provided they are properly interpreted, if not orally at least mentally, but taken literally are absurd. The first expression means, of course, that a quantity having a value of two units of a certain kind

is to be taken four times; and similarly 8÷4 means that a total quantity of a certain kind is measured by four units or by two units of the same kind. Of course, in all mathematical calculations we ultimately operate with pure symbols, and the operations do not affect the unit of measure; but in the beginning we should make constant reference to measured quantity, and always should be prepared to interpret the symbols and the processes.

3. Number, then, as distinct from the magnitude which is the unit of reference, and from the magnitude which is the unity or limited quality to be measured, is:

The repetition of a certain magnitude used as the unit of measurement to equal or express the comparative value of a magnitude of the same kind. It always answers the question "How many ?"

This "how many" may assume two related aspects: either how many times one part as unit has to be taken or repeated to make up the whole quantity; or how many parts as units, each taken once, compose the whole. In the first case, the times of repetition of the measuring unit is mentally the more prominent; in the second, the actual number of measuring parts; e. g., in thinking of forty yards, we may at one time dwell on the forty times the unit is repeated; at another time, on the actual forty parts making the unified whole.

As already said, the number and the measuring unit together give the absolute magnitude of the quantity. The number by itself indicates its relative value. It always expresses ratio*-i. e., the relation which the

* Hence, again, the absurdity of multiplying pure number or dividing it into parts. We may divide a ratio, but not into parts.

magnitude to be measured bears to the unit of reference. Seven, as pure number, expresses equally the ratio of 1 foot to 7 feet, of 1 inch to 7 inches, of 1 day to 1 week, of $1,000 to $7,000, and so on indefinitely. Simply as seven it has no meaning, no definite value at all; it only states a possible measurement.

This definition arrived at from psychological analysis is that given by some of the greatest mathematicians on a strictly mathematical basis, as may be seen from comparison with the following definitions :

Newton's.-Number is the abstract ratio of one quantity to another quantity of the same kind.

Euler's.-Number is the ratio of one quantity to another quantity taken as unit.*

PHASES OF NUMBER.-The aspects of number follow directly from what has been said. Quantity, the unity measured, whether a "collection of objects" or a physical whole, is continuous, an undefined how much; number as measuring value is discrete, how many. The magnitude, muchness, before measurement is mere unity; after measurement it is a sum taken as an integer that is, an aggregation of parts (units) making up one whole; number as showing how many refers to the units, which put together make the sum. Quantity, measured magnitude, is always concrete; it is a certain kind of magnitude, length, volume, weight, area, amount of cost,

* J. C. Gláshan, one of the acutest of living mathematicians, defines thus: "A unit is any standard of reference employed in counting any collection of objects, or in measuring any magnitude. A number is that which is applied to a unit to express the comparative magnitude of a quantity of the same kind as the unit." (See his Arithmetic for High Schools, etc.)

etc.

"Number," as simply defining the how many units of measurement, is always abstract.

The conception of measuring parts and of times of repetition is inseparable from number as expressing the numerical value of a quantity; as discrete, it is so many parts taken one time-constituting the unity; as abstract, it is one part taken so many times. In the one case, as before suggested, attention is more upon the numbered parts, in the other, more upon the number of the parts. They are absolutely correlative conceptions of the same measured magnitude. That is, a value of $50 may be regarded as determined by taking $1 fifty times, or by taking $50—that is, a whole of fifty parts -one time. The numerical process and the resulting numerical value are the same, however we arrive at the number-i. e., the ratio of measured quantity to measuring unit. As this conception of the relation between parts and times in the measurement of quantity is essential to the interpretation of numerical operations, we may give it a little further consideration.

We wish to know the amount of money in a roll of dollar bills. We take five dollars, say, as a convenient measuring unit; we separate our undefined whole into groups of five dollars each; we count these groups and find that there are ten of them-i. e., the numerical value is ten; we have now a definite idea both of the measuring unit and of the times it is repeated, and so have reached a definite idea of the amount of money in the roll of bills. We began with a vague whole, an undefined unity; we broke it up into parts (analysis), and by relating (counting) the parts we arrived at our unity again; the same unity, yet not the same as regards the

attitude of the mind towards it. It is now a definite unity constituted by a known number of definite parts; it is a sum of units. On the analytic side of this defining process the emphasis is on the parts, the units; on the synthetic side the emphasis is on the defined unity, the sum. The parts are means to an end; they exist only for the sake of the end, the sum. The ten units-that is, the unit repeated ten times—make up, are, the one sum-i. e., the sum taken one time.

Further, since the unit of measure is itself measured by' a smaller unit, the dollar, the same psychological explanation applies to the measurement of the quantity by means of this smaller unit. The five-dollar unit taken ten times is identical with ten of these units taken once. We are conscious, also, that any part of this five-dollar unit taken ten times is identical with ten such parts taken once. That is, $1, taken ten times, is a whole of $10 taken once; and since this is true of every dollar in the five, our measurement gives a whole of $10 taken once, a whole of $10 taken twice, and so on; that is, altogether, a whole of $10 taken five times. In other words, the measurement, ten groups (or units) of five dollars each necessarily implies the correlative measurement, five groups of ten dollars each.

This rhythmic process of parting and wholing which leads to all definite quantitative ideas, and involves the correlation of times and parts, may be illustrated by simple intuitions. In measuring a certain length we find it, let us suppose, to contain four parts of three feet each; then the relation between parts (measuring units) and numerical value (times of repetition) may be