we have in all"; or the teacher might ask, at the end of every problem, "What were we counting up or measuring that time?" letting each one interpret as he pleased. Just how far this is carried is a matter of detail; what is not a matter of detail is that the habit of interpretation be formed by continually referring the numbers to some quantity.

2. The unit is never to be taught as a fixed thing (e. g., as in the Grube method), but always as a unit of measurement. One is never one thing simply, but always that one thing used as a basis for counting off and thus measuring some whole or quantity. Absolutely everything and anything which we attend to is one; is made one by the very act of attending. If we could take in the whole system of things in one observation, that would be one; if we could isolate an atom and look at that, it would equally be one. The forest is one when we view it as a whole; the tree, the branch, the stem, the leaf, the cell in the leaf, is equally one when it becomes the object or whole with which we are occupied. But this oneness, this unity possessed by every object of attention, has nothing but the name in common with the numerical unit. In itself it is not quantitative at all; it is mere unity of quality, of meaning. It becomes a quantitative unity (a quantity or magnitude) only when considered as limited (page 36), and as an end to be reached by the use of certain means. It becomes a unit only when used as one of the means to construct a value equivalent to a certain other value. The assumption that some one object is the natural unit of quantity, which is then increased by bringing in other objects, is the very opposite of the truth; number does

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not arise at all until we cease taking objects as objects, and regard them simply as parts which make up a whole, as units which measure a magnitude (see pages 24, 42, on Abstraction). It is perfectly clear, therefore, that the method of "close observation" of objects is essentially vicious; what is claimed as its merit is in reality a grave defect. The child, according to the advocates of this method, "sees what is brought to his notice and sees all about it." But in seeing all about the things there must be neglect of the numerical abstraction which sees nothing about the things save this alone they are parts of one whole. There may be a discriminating and relating of qualities which give the things individual meaning; but there is not the process—at least the process is impeded-which constructs quantitative units into a defined quantitative whole.

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This is plainly so in case of units like dollars, inches, pounds, minutes, etc. They are units not in virtue of any quality absolutely inherent in them, but in virtue of their use in measuring cost, length, weight, duration, etc. It may be said that this is not so in the case of books, apples, boys, etc.; that here each book, apple, etc., is a unit in itself. But this is to fall into the error of separating counting from measuring, already referred to. The book, the crayon, or the cube, is a unity, a whole, in itself, but it is not a unit save as used to count up (value) the total amount. The only point is that this counting gives very crude measurement. The unit, book, pie, and so on, is not itself measured by minor units of the same kind. We are measuring with an unmeasured unit, and so the result of our measurement is exceedingly vague and inaccurate, just as it would be

to measure length by steps which had themselves no definite length; cost of other goods by potatoes themselves changing in value, etc.*

Further, since the measuring unit is itself measured,. is itself made up of minor or sub-units, it may be of any numerical value, denoting two, three, four, five, six, etc., of such minor units. Twenty-five pounds taken as a basis of measurement is a unit, is one; taken with a reference to the minor unit (1 lb.), by which itself is measured, it is a defined unity, a sum. Thirty-six feet referred to a measuring quantity of three feet has a numerical value twelve; and the three feet is just as much a unit (one, or 1) as one foot is in measuring 12 feet. So the quantities 9 piles of silver coin of ten dollars each, and 25 pages of thirty lines each, have respectively the numerical values of 9 and 25-that is, nine units of measurement ("ones") and twenty-five units ("ones") of measurement. This is the very basis of our system of notation. The numerical value, hundred, applied to any measuring unit, denotes a quantity consisting of 10 ten-units; the number, thousand, measures a quantity which is composed of 10 hundred-units; the number, tenth, applied to any unit, measures that quantity which taken ten times makes up the unit of reference; the number, hundredth, used with any measuring unit denotes that quantity which taken ten times makes up one tenth of the unit of reference, etc.

* Hence, once more, the fallacious ideas introduced by our arithmetics in illustrating so much by these unmeasured units partially qualitative and only partially quantitative-the pencil, the apple, the orange, and the universal pie-and so little by the definite units of length, size, weight, money value, etc.

3. The true method, then, may be summarised by saying that the proper introduction to numerical operations is by presenting the material in such a way as to require a mental operation of rhythmic parting and wholing—that is, a quality or magnitude is to be presented in such a way as to involve both separation (mental separation, that is, of values, not necessarily physical partition) into parts and the recomposition of the parts into the whole. The analysis gives possession of the unit of measurement; the synthesis, or recomposition, gives the absolute value of the magnitude; the process itself brings out the ratio, the pure number.

We thus see the fundamental fallacy of the Grube method in another light. Just as, upon the whole, it proceeds from the mere observation of objects instead of from the constructive use of them, so it works with fixed units instead of with a whole quantity which is measured by the application of a unit of measurement. The superiority of the Grube method to some of the other methods, both in the way of introducing objects instead of dealing merely with numerical symbols, and in the way of systematic and definite instead of haphazard and vague work, has tended to blind educators to its fundamentally bad character, psychologically speaking. There is no need to dwell upon this at length after the previous exposition, but the following points may be noted:

(a) In proceeding from one to two, then to three, etc., it leaves out of sight the principle of limit, which is both mathematically and psychologically fundamental. There is no limited quality, no magnitude, with its own intrinsic unity, which sets bounds to and gives the rea

son for the numerical operation. Number is separated from its reason, its function, measurement of quantity, and so becomes meaningless and mechanical. There is no inner need, no felt necessity, for performing the operations with number. They are artificial. We are dealing with parts which refer to no whole, with units which do not refer to a magnitude. It is as sensible as it would be to make a child learn all the various parts of a machine, and carefully conceal from him the purpose of the machine-what it is for, what it does—and thus make the existence of the parts wholly unintelligible.

(b) In beginning with the fixed unit one object (1), then going on to two objects, three objects, then other fixed units, there is no intrinsic psychological connection among the various operations. We may add, we may subtract, we may find a ratio; but addition, subtraction, ratio, remain (psychologically) separate processes. According to true psychology, we begin with a whole of quantity, which on one side is analysed into its units of measurement, while on the other these units are synthesised to constitute the value of the original magnitude; we have parts which refer to a whole, and units which make a sum. Here the addition and subtraction are psychological counterparts; we actually perform both these operations, whether we consciously note more than one of them or not. Similarly, we go through a process of ratio-ing in the rhythmic construction of the whole (much) out of the units (many); the conscious grasp of the principle of ratio will therefore involve no new operation, but simply reflection upon what we have already done. First one process, then another, then another, and so on, is the law of the Grube method