measuring units-drachms, ounces, etc.; in the first type of measuring we have not such a scale.

Finally, the pound itself may be defined not only as 16 ounces, but also as bearing a relation to some other standard; as, e. g., a cubic foot of distilled water at the temperature of 39.83 weighs 62 pounds, the linear foot itself being defined as a definite part of a pendulum which, under given conditions, vibrates seconds in a given latitude (see page 46).

The Specific Numerical Operations.—The fundamental operations, as already said, are phases in the development of the measuring process.

1. We have seen that the comparison of two quantities in order to select the one fittest for a given end not only gives rise to quantitative ideas, but also tends to make them more clear and definite. Each of the quantities is at first a vague whole; but one is longer or shorter, heavier or lighter, in a word, more or less than the other. Here we have the germinal idea of addition and subtraction. The difference between the quantities will be a vague muchness, just as the quantities themselves are vague, and will become better defined just as these become better defined. This better definition arises with the first stage of measurementthat of the undefined unit. We begin with measuring a collection of objects by counting them off, and this suggests the measuring of a continuous quantity in a similar way—that is, by counting it off in so many paces, hand-breadths, etc. Now, in the use of the inexact unit there is given a more definite idea of the quantities and of the more or less which distinguishes them, but no explicit thought of the ratio of one to the other;

there is a counting of like things but not of equal things. In other words, the process of counting with an unmeasured unit gives us arithmetically Addition and Subtraction. The result is definite simply as to more or less of magnitude. It shows how many more coins there are in one heap than in another, how many more paces in one distance than another, in, etc. It gives an idea of the relative value in this one point of moreness or aggregation, but it does not bring into consciousness what multiple, or part, one of the quantities is of the other, or of their difference. This is a more complex conception, and so a later mental product.

2. With the development of the idea of quantity in fulness and accuracy the second stage of measurement is reached, in which the measuring unit is uniform and defined in terms homogeneous with the measured quantity.

This principle of measuring with an exact uniti. e., a unit which is itself made up of minor units in the same scale-gives rise to Multiplication and Division, and is in reality the principle of ratio. In the addition or subtraction of two quantities we are not conscious of their ratio; we do not even use the idea of their ratio. In multiplication and division we are constantly dealing with ratio. We do not discover merely that one quantity is more or less than another, but that one is a certain part or multiple of another. When, for example, we multiply $4 by 5 we are using ratio; we have a sum of money measured by 5 units of $4 each, where the number 5 is the ratio of the quantity measured to the measuring unit. In division, the inverse of multiplication, ratio is still more prominent.

The idea of ratio involved in multiplication and division is a much more practical one than that of mere aggregation (more or less) involved in addition and subtraction, because it helps to a more accurate adjustment of means to end. Suppose a man in receipt of a certain salary knew that the rent of one of two houses is $100 a year more than that of the other, but could not tell the ratio of the $100 to his salary, it is obvious that he would have but little to guide him to a decision. But if he knows that $100 is one fifth or one fiftieth of his entire income, he has clear and positive knowledge for his guidance.

With ratio multiplication and division-go the simpler forms and processes of fractions.*

3. The principle of measuring one scale in terms of another gives us arithmetically proportion, and the operations involving it, such as percentage and multiplication and division of fractions, and brings out the idea of the equation.

THE ORDER OF ARITHMETICAL INSTRUCTION.-We have already seen one fundamental objection to the ordinary method of teaching number, whether as carried on in a haphazard way or by what is known as "the Grube" method; it takes number to be a fixed

* In external form, but not in internal meaning, other fractions belong here also. For example, the ratio of 14 to 3 may be written 14 ÷ 3 or 4; in any case, the idea is to discover how many units of the value of 3 measure the value of 14 units, but the very fact that 3 is taken as the unit shows the meaning to be the discovery of the ratio of 14 to 3 as unity. Whether the result can actually be written in integral form or not is of no consequence in principle, so long as the process is the attempt to discover the ratio to unity; the process is of 14.

quantity, instead of a mental operation concerned in measuring quantity. We can now appreciate another fundamental objection: it attempts to teach all the operations simultaneously, and thus neglects the fact of growth in psychological complexity corresponding to the development of the stages of measurement. It takes each number as an entity in itself, and exhausts all the operations (except formal proportion *) that can be performed within the range of that number. It assumes that the logical order is the order of growth in psychological difficulty. All operations are implied even in counting, but are not therefore identical.

Logically, or as processes, all operations are implied, even in counting. To count up a total of four apples involves multiplication and division, and thus ratio and fractions. When we have counted 3 of the 4 apples, we have taken a first 1, a second 1, and a third 1— that is, a total of three 1's-out of the 4 which compose the original quantity. We have divided the original quantity of apples into partial values as units, and have taken one of those units so many times; this is multiplication. But it does not follow that, because the operations are logically implied in this process, they are therefore the same in their complete development and all equal in point of psychological difficulty; much less that they should be definitely evolved in consciousness and all taught together. The acorn implies the oak, but the oak is not the acorn. Multiplication is im

* Why not proportion, or even logarithms, on the principle that everything that is logically correlative should be taught at once? The logarithm is just as much involved in say 8, as are all the multiplications and additions which can be deduced from it.

plied in the simple act of counting, and has its genesis in addition; but multiplication is not merely counting, nor is it identical with addition. The operation indicated in $2 + $2 + $2 + $2 = $8 may be performed, and in the initial stages of mental growth is performed, without the conscious recognition that eight is four times two. The latter is implied in the former, and in due time is evolved from it; but for this very reason it is a later and more complex conception, and therefore makes a severer demand upon conscious attention. The summing process is made comparatively easy through the use of objects; it is little more than the perception of related things. The multiplication process is more complex, because it demands the actual use and more or less conscious grasp of ratio, or times, the abstract element of all numbers; it is the conception of the relation of things. We might go on adding twos, or threes, or fours, instantly merging each successive addend in the growing aggregate, and, never returning to the addends, correctly obtain the respective sums without the more abstract conception of times ever arising—that is, without ever being conscious that the "sum" is a product of which the times of repetition of the addend is one of the factors. Certainly this more abstract notion does not arise at first in the development of numerical ideas in either the child or the race.

If anyone still maintains that addition (of equal addends) and multiplication are identical processes, let him prove by mere summing (or counting) that the square root of two, multiplied by the square root of three, is equal to the square root of six; or find by logarithms the sum of a given number of equal addends.