At this point in the American textbooks it is the custom to insert a brief treatment of measurement, explaining what is meant by ratio, commensurable and incommensurable quantities, constant and variable, and limit, and introducing one or more propositions relating to limits. The object of this departure from the ancient sequence, which postponed this subject to the book on ratio and proportion, is to treat the circle more completely in Book III. It must be confessed that the treatment is not as scientific as that of Euclid, as will be explained under Book III, but it is far better suited to the mind of a boy or girl. It begins by defining measurement in a practical way, as the finding of the number of times a quantity of any kind contains a known quantity of the same kind. Of course this gives a number, but this number may be a surd, like [sqrt]2. In other words, the magnitude measured may be incommensurable with the unit of measure, a seeming paradox. With this difficulty, however, the pupil should not be called upon to contend at this stage in his progress. The whole subject of incommensurables might safely be postponed, although it may be treated in an elementary fashion at this time. The fact that the measure of the diagonal of a square, of which a side is unity, is [sqrt]2, and that this measure is an incommensurable number, is not so paradoxical as it seems, the paradox being verbal rather than actual. It is then customary to define ratio as the quotient of the numerical measures of two quantities in terms of a common unit. This brings all ratios to the basis of numerical fractions, and while it is not scientifically so satisfactory as the ancient concept which considered the terms as lines, surfaces, angles, or solids, it is more practical, and it suffices for the needs of elementary pupils. "Commensurable," "incommensurable," "constant," and "variable" are then defined, and these definitions are followed by a brief discussion of limit. It simplifies the treatment of this subject to state at once that there are two classes of limits,--those which the variable actually reaches, and those which it can only approach indefinitely near. We find the one as frequently as we find the other, although it is the latter that is referred to in geometry. For example, the superior limit of a chord is a diameter, and this limit the chord may reach. The inferior limit is zero, but we do not consider the chord as reaching this limit. It is also well to call the attention of pupils to the fact that a quantity may decrease towards its limit as well as increase towards it.