means of visible illustrations. The different objects which have been mentioned for counting in whole numbers may equally serve for imparting to young people the first notions of fractions. A number of such objects, being considered as a whole and variously divided into equal parts or fractional numbers, would, by the addition and subdivision of these, illustrate the relative value and the elementary operations of simple fractions. This, however, may perhaps be still better effected by the following contrivance :—

Let about 16 or 18 thin slips of wood or pasteboard, about half an inch in breadth, be made all exactly the same length, say one foot. (This length is convenient, and will, besides, accustom the eye of the child to a useful measure.) Let them be divided by a line across the breadth, the first into two equal portions, the second into three, the the third into four, and so on up to the eleventh, which will be composed of twelve equal parts; a few other slips may be respectively divided into 15, 18, 20, 24, 36, 48, 60, 72, and 84 equal parts, which numbers are chosen on account of their having a great number of divisors. Let the lines indicating different subdivisions be of different colors, and those indicating equal portions in the different slips be of the same color-all the halves throughout being thus of one color, all the thirds of another, and so on. Let also the denominator, that is, the number of parts into which the foot-slips are divided, be marked at one of the ends of each slip. These colored lines and written denominators will greatly assist in distinguishing at once the different fractions, reducing them to their lowest terins, and finding out their common denominator.

The pupil with these slips placed side by side under his eye, should be called upon to observe the various subdivisions of the foot which are marked on them, and be told the names by which are denominated the equal parts of each slip, halves, thirds, fourths or quarters, &c.; he may, from these, discover by analogy, the names of the others. He should be made successively to notice that 2, 3, 4, &c., are equal to one another; that, 2, 3, &c., are the same; that, is greater than, greater than , &c.; that are less than 4, 3 less than ; &c.; that the fraction is greater in proportion as the numerator is increased, or the denominator lessened, and vice versa. He should add, subtract, find a common denominator, and reduce fractions to their lowest terms. In short, he might, by means of this simple apparatus, and, under the guidance of a judicious teacher, gain a clear acquaintance with the denominations, nature, value, and properties of common fractions, long before he could safely be introduced to their numerical symbols and to their abstract forms.

4. Forms; Geometrical solids; Architectural game.

In order promptly to familiarize the pupil with the most general forms and the terms expressive of them, a collection of small geometrical solids should he exhibited to him, such as spheres, cylinders, cones, prisms, pyramids, and the regular geometrical bodies in different dimensions, as also a cone with its several sections. In minutely examining each of these, his attention may easily be directed, by a natural analysis, from the solids to the surfaces, triangles, quadrilaterals, and polygons; from these to the angles, lines, and points. In comparing them afterwards, he may find out himself their differences, and classify them; and, in stating the result of his examination, he is led to the use and to the definition of the scientific terms which designate them, and to the consideration of the first elements of geometry.

By a reference to the geometrical solids a child may easily understand what is meant by vertical and horizontal; perpendicular and oblique; parallel and divergent, and convergent; right, acute, and obtuse angles; circle, circumference, and diameter; he may be shown. the principal properties of triangles, the mode of measuring and dividing angles, the relative length of circumference and diameter, and may be taught by means of small square blocks or cubes, how to measure rectangular superficies and solids.

If the child be made to sketch the outlines of these solids, it will be a further preparation for his future study of that science; for these diagram sketches, within the power of a young child-and his first step in the useful practice of drawing from nature, will direct his attention more closely to the geometrical forms, will familiarize him with the terms and graphic representations of them, and will give him some practical notions of perspective. The precision and accuracy of eye, gained, at the same time, by the habit of drawing, would considerably assist him in clearly conceiving the forms, proportions, and dimensions of objects. The facility and correctness, also, with which he will execute these figures, if he has early practiced drawing, will, at a future period, render geometry much more attractive; whilst the elements of this science will, in their turn, tend to give a useful direction to linear drawing.

The practice of ascertaining the various parts, substances, colors, and forms of objects, is an effectual preparation for the study of the natural sciences; it can not fail to impart accuracy and acuteness to the perceptive powers of young persons; it will accustom them to observe and analyze things minutely; while all the

terms relative to these different points will considerably extend their Vocabulary.

To those who advocate for children science in play, we will suggest that the young mind may be effectually familiarized with forms and proportions by means of an architectural game composed of brickshaped pieces, and others in imitation of those which enter into the construction of buildings-blocks of different sizes (say, from one inch to four inches in length, one inch in breadth, and half an inch in thickness,) cubes, arches, columns, with detached bases, capitals, and moldings, in different orders of architecture. These building. materials may be so contrived as to present, by their various combinations, illustrations of geometrical propositions, and, by their superstructure, edifices in different styles of architecture. They should consist of close-grained wood, of two contrasting colors, so as to please the eye by their neatness and symmetrical arrangements; and if they be made with mathematical accuracy, and on a scale founded on the national measures, they will be easily raised in conformity to any architectural design, while the eye will be early habituated to a useful measure. The author, anxious to give his children the benefit of such a game, has constructed one with box and Brazil wood (white and red,) composed of about six hundred pieces of various sizes and geometrical forms, on the above-mentioned scale of measurement. It has been for his young family not only an exhaustless source of pleasure and instruction, but an efficient means of forming habits of patience and enticing them to efforts of invention.

SECT. 11.-EXERCISES IN OBSERVATION.

1. Properties, Comparisons, and Classification of objects. From the age of eight or nine, when the child's perceptive faculties have been exercised on the most apparent properties of things, and when he has learned to confine and prolong his attention, he should be required to examine objects more minutely, to compare them under different points of view, and to state in what particular two or more resemble or differ. These exercises would prove highly interesting to young people, who delight in discovering differences. between similar things, and resemblances between different things. The judgment, according to Locke, is exercised by the first act, and the imagination by the second: all the intellectual powers, in fact, which have comparison for their basis, would be thus highly cultivated. He who is best able to compare will know best how to analyze, to abstract, to generalize, to classify, to judge-in one word,

to reason.

Various objects should be successively submitted to the organs of sense, and the relations in which they stand to each other be duly examined, in order that, by observation and comparison, their particular properties may be discovered, as well those which are relative to our constitution as those which are inherent in the objects themselves. A true knowledge of things consists in a perfect acquaintance with all their properties. When objects have been considered in all their bearings, the child may be directed how to classify them according to the similarity of their essential attributes. It is, in fact, the relation of resemblance which, by the general notions and corresponding general terms that flow from it, becomes the source of classification and definition, and of all that is valuable in language.

As the attributes inherent in matter may not all present themselves to the mind of the teacher at the very moment when he wishes to direct the attention of the pupil to them, tables containing in juxtaposition adjectives of opposite meanings would enable him to point out all the properties the presence or absence of which can be ascertained in objects.

Every new discovery which results from the investigation of objects exercises the understanding, leads to a knowledge of the true essence of things, and stores the memory with adjectives and abstract nouns, the chief materials of descriptive and philosophical language. A familiarity with such terms, by generating a habit of nice discrimination, and enriching the imagination with vivid conceptions of things, constitutes the characteristic elements of eloquence. Uneducated people are particularly deficient in these two species of words. The child being also led to distinguish the properties which are natural or artificial, essential or accidental, permanent or transient, absolute or relative, and to discover those which belong to one object exclusively, or are common to several, will find no difficulty in making classifications, or availing himself of those already existing, and of their corresponding nomenclatures. Classification is the indispensable complement of observation.

As young persons collect facts, they must be frequently exercised in classifying them with reference to their resemblance or difference. If any number of objects is considered with regard to one or several points of resemblance, the collection constitutes a class named genus; subdivisions of these into classes of objects having properties in common and distinct from the rest, form as many species; finally, when, on a closer examination, single objects are considered in reference to properties which are peculiar to them, they are denominated individuals. The child must be shown that the terms genus and species

are relative: the same class which is a genus with reference to the sub-classes, or species included in it, may be itself a species relatively to a more extensive, or, as it is often called, a superior genus. Bird, for example, a genus with regard to the different species eagle, sparrow, &c., is, in its turn, a species of the genus animal, which is itself a species with respect to the superior genus organized being. Filial love is a species of the genus affection; affection, a species of the genus goodness; and goodness, a species of the genus inclination. The distinction of generic and specific terms applies to a very extensive range of mental conceptions.

The complex operation of classifying things according to their points of resemblance, and of distinguishing them by their points of dissimilarity, is one of the highest exercises of our reason and the most admirable effect of analysis. It will develop in a child the powers of observation, abstraction, and generalization, and will prepare him for the study of the natural and experimental sciences, by giving him habits of inductive reasoning-a principle on which these sciences rest.

Nothing is more beneficial to the mind than the early habit of referring particular ideas to general principles, and classifying objects and the notions acquired about them. The memory will best retain the information intrusted to its keeping when arranged according to some principle of generalization. Classification leads to the clear. conception and exact definition of terms; because the names given to our generalizations in order to classify things, are connected in the mind with the peculiarities that characterize these things it becomes the more useful as ideas accumulate on the mind; for, in general, confusion does not arise so much from the number of ideas, as from the incapability of conceiving them clearly and arranging them in a proper order. Classification is the ground-work of inductive philosophy, and of all scientific investigations.

2. Incidental investigations about Objects.

The act of observing, which springs from the natural desire for knowledge, reacts on that desire and stimulates it, when it has become a habit: if, therefore, the child's powers of observation have been judiciously exercised, his inquisitiveness will increase with his mental development. He may then gradually be brought to investigate incidents connected with an object: among others, what are its different uses, the country whence it comes, the mode of production, the process of fabrication, the instruments employed in making it, and the trades concurring to its completion.